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Mathematics LibreTexts

10-1. The Ellipse

The Ellipse
In this section, you will:
  • Write equations of ellipses in standard form.
  • Graph ellipses centered at the origin.
  • Graph ellipses not centered at the origin.
  • Solve applied problems involving ellipses.
<figure class="small" id="Figure_10_01_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)</figcaption> </figure>

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in [link], is such a room.1 It is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in [link].

<figure id="Figure_10_01_002"></figure>

Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See [link].

<figure class="small" id="Figure_10_01_003"></figure>

Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is aco-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See [link].

<figure class="medium" id="Figure_10_01_004"></figure>

In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.

To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

Deriving the Equation of an Ellipse Centered at the Origin

To derive the equation of an ellipse centered at the origin, we begin with the foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −c,0 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The ellipse is the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) such that the sum of the distances from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) to the foci is constant, as shown in [link].

<figure class="small" id="Figure_10_01_014"></figure>

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a,0 ) is a vertex of the ellipse, the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −c,0 ) to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> c,0 ) to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a,0 ) is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>−</mo><mi>c</mi></mrow></annotation-xml></semantics></math>. The sum of the distances from the foci to the vertex is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a+c )+( a−c )=2a

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) is a point on the ellipse, then we can define the following variables:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>d</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 =the distance from (−c,0) to (x,y) d 2 =the distance from (c,0) to (x,y)

By the definition of an ellipse,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> d 1 + d 2  is constant for any point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) on the ellipse. We know that the sum of these distances is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>for the vertex<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>It follows that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> d 1 + d 2 =2a for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>                                      </mtext><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> d 1 + d 2 = (x−(−c)) 2 + (y−0) 2 + (x−c) 2 + (y−0) 2 =2a Distance formula (x+c) 2 + y 2 + (x−c) 2 +y 2 =2a Simplify expressions.                              (x+c) 2 + y 2 =2a− (x−c) 2 + y 2 Move radical to opposite side.                               (x+c) 2 + y 2 = [ 2a− (x−c) 2 + y 2 ] 2 Square both sides.                      x 2 +2cx+ c 2 + y 2 =4 a 2 −4a (x−c) 2 + y 2 + (x−c) 2 + y 2 Expand the squares.                      x 2 +2cx+ c 2 + y 2 =4 a 2 −4a (x−c) 2 + y 2 + x 2 −2cx+ c 2 +y 2 Expand remaining squares.                                              2cx=4 a 2 −4a (x−c) 2 + y 2 −2cx Combine like terms.                                   4cx−4 a 2 =−4a (x−c) 2 + y 2 Isolate the radical.                                        cx− a 2 =−a (x−c) 2 + y 2Divide by 4.                                     [ cx− a 2 ] 2 = a 2 [ (x−c) 2 + y 2 ] 2 Square both sides.                      c 2 x 2 −2 a 2 cx+ a4 = a 2 ( x 2 −2cx+ c 2 + y 2 ) Expand the squares.                      c 2 x 2 −2 a 2 cx+ a 4 = a 2 x 2 −2 a 2 cx+ a 2 c 2 + a 2 y 2Distribute  a 2 .                   a 2 x 2 − c 2 x 2 + a 2 y 2 = a 4 − a 2 c 2 Rewrite.                     x 2 ( a 2 − c 2 )+ a 2 y 2 = a 2 ( a 2− c 2 ) Factor common terms.                                x 2 b 2 + a 2 y 2 = a 2 b 2 Set  b 2 = a 2 − c 2 .                              x 2 b 2 a 2b 2 + a 2 y 2 a 2 b 2 = a 2 b 2 a 2 b 2 Divide both sides by  a 2 b 2 .                                       x 2 a 2 + y 2 b 2 =1 Simplify.

Thus, the standard equation of an ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1.This equation defines an ellipse centered at the origin. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>,</mo></mrow></annotation-xml></semantics></math>the ellipse is stretched further in the horizontal direction, and if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>></mo><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the ellipse is stretched further in the vertical direction.

Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.

The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

Standard Forms of the Equation of an Ellipse with Center (0,0)

The standard form of the equation of an ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and major axis on the x-axis is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 a 2 + y 2 b 2 =1

where

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the length of the major axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±a,0 )
  • the length of the minor axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±b )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±c,0 ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . See [link]a

The standard form of the equation of an ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and major axis on the y-axis is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 b 2 + y 2 a 2 =1

where

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the length of the major axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±a )
  • the length of the minor axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±b,0 )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±c ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . See [link]b

Note that the vertices, co-vertices, and foci are related by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.

<figure data-label="" id="Figure_10_01_005"> <figcaption>(a) Horizontal ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) (b) Vertical ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 )</figcaption> </figure>

Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.

  1. Determine whether the major axis lies on the x- or y-axis.
    1. If the given coordinates of the vertices and foci have the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±a,0 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>±</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>respectively, then the major axis is the x-axis. Use the standard form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1.
    2. If the given coordinates of the vertices and foci have the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±a ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>±</mo><mi>c</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math>respectively, then the major axis is the y-axis. Use the standard form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 b 2 + y 2 a 2 =1.
  2. Use the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 , along with the given coordinates of the vertices and foci, to solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 .
  3. Substitute the values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2  into the standard form of the equation determined in Step 1.
Writing the Equation of an Ellipse Centered at the Origin in Standard Form

What is the standard form equation of the ellipse that has vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±8,0 ) and foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±5,0 )? 

The foci are on the x-axis, so the major axis is the x-axis. Thus, the equation will have the form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 a 2 + y 2 b 2 =1

The vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±8,0 ),so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>=</mo><mn>8</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =64.

The foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±5,0 ),so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>=</mo><mn>5</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 =25.

We know that the vertices and foci are related by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 , we have:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>c</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 = a 2 − b 2 25=64− b 2 Substitute for  c 2  and  a 2 . b 2 =39 Solve for  b 2 .

Now we need only substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =64 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =39  into the standard form of the equation. The equation of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 64 + y 2 39 =1.

What is the standard form equation of the ellipse that has vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±4 ) and foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,± 15 )?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 16 =1

Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?

Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±a,0 ) or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mo>±</mo><mi>a</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></annotation-xml></semantics></math>Similarly, the coordinates of the foci will always have the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±c,0 ) or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mo>±</mo><mi>c</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></annotation-xml></semantics></math>Knowing this, we can use<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>from the given points, along with the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 ,to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 .

Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units horizontally and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units vertically, the center of the ellipse will be<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ). This translation results in the standard form of the equation we saw previously, with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>replaced by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−h ) and y replaced by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> y−k ).

Standard Forms of the Equation of an Ellipse with Center (h, k)

The standard form of the equation of an ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h, k ) and major axis parallel to the x-axis is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−h ) 2 a 2 + ( y−k ) 2 b 2 =1

where

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the length of the major axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±a,k )
  • the length of the minor axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±b )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±c,k ),where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . See [link]a

The standard form of the equation of an ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ) and major axis parallel to the y-axis is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−h ) 2 b 2 + ( y−k ) 2 a 2 =1

where

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the length of the major axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±a )
  • the length of the minor axis is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi></mrow></annotation-xml></semantics></math>
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±b,k )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±c ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . See [link]b

Just as with ellipses centered at the origin, ellipses that are centered at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ) have vertices, co-vertices, and foci that are related by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.

<figure id="Figure_10_01_006"> <figcaption>(a) Horizontal ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ) (b) Vertical ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k )</figcaption> </figure>

Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.

  1. Determine whether the major axis is parallel to the x- or y-axis.
    1. If the y-coordinates of the given vertices and foci are the same, then the major axis is parallel to the x-axis. Use the standard form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1.
    2. If the x-coordinates of the given vertices and foci are the same, then the major axis is parallel to the y-axis. Use the standard form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1.
  2. Identify the center of the ellipse<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ) using the midpoint formula and the given coordinates for the vertices.
  3. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2  by solving for the length of the major axis,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math> which is the distance between the given vertices.
  4. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2  using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>,</mo></mrow></annotation-xml></semantics></math> found in Step 2, along with the given coordinates for the foci.
  5. Solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2  using the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 .
  6. Substitute the values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>, </mo><mi>k</mi><mo>, </mo><msup><mi>a</mi></msup></mrow></annotation-xml></semantics></math> 2 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2  into the standard form of the equation determined in Step 1.
Writing the Equation of an Ellipse Centered at a Point Other Than the Origin

What is the standard form equation of the ellipse that has vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−8 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,2 )

and foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−7 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,1 )?

The x-coordinates of the vertices and foci are the same, so the major axis is parallel to the y-axis. Thus, the equation of the ellipse will have the form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−h ) 2 b 2 + ( y−k ) 2 a 2 =1

First, we identify the center,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ). The center is halfway between the vertices,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−8 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,2 ). Applying the midpoint formula, we have:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −2+(−2) 2 , −8+2 2 )          =(−2,−3)

Next, we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 . The length of the major axis,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math> is bounded by the vertices. We solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by finding the distance between the y-coordinates of the vertices.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mi>a</mi><mo>=</mo><mn>2</mn><mo>−</mo><mo stretchy="false">(</mo><mn>−8</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2a=10 a=5

So<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =25.

Now we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 . The foci are given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±c ). So,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k−c )=( −2,−7 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k+c )=( −2,1 ). We substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>−3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>using either of these points to solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>k</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −3+c=1               c=4

So<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 =16.

Next, we solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2  using the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2               16=25− b 2 b 2 =9

Finally, we substitute the values found for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>,</mo><mi>k</mi><mo>,</mo><msup/></mrow></annotation-xml></semantics></math> a 2 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2  into the standard form equation for an ellipse:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x+2 ) 2 9 + ( y+3 ) 2 25 =1

What is the standard form equation of the ellipse that has vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,3 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,3 ) and foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1−2 3 ,3 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1+2 3 ,3 )?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−1 ) 2 16 + ( y−3 ) 2 4 =1

Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1, a>b for horizontal ellipses and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 b 2 + y 2 a 2 =1, a>b for vertical ellipses.

Given the standard form of an equation for an ellipse centered at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 0 ), sketch the graph.

  1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
    1. If the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1,  where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then
      • the major axis is the x-axis
      • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±a,0 )
      • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±b )
      • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±c,0 )
    2. If the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 b 2 + y 2 a 2 =1,where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then
      • the major axis is the y-axis
      • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±a )
      • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±b,0 )
      • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±c )
  2. Solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>using the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 .
  3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an Ellipse Centered at the Origin

Graph the ellipse given by the equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 9 + y 2 25 =1. Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>25</mn><mo>></mo><mn>9</mn><mo>,</mo></mrow></annotation-xml></semantics></math>the major axis is on the y-axis. Therefore, the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 b 2 + y 2 a 2 =1,where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =9 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =25. It follows that:

the center of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 )
the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±a )=( 0,± 25 )=( 0,±5 )
the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±b,0 )=( ± 9 ,0 )=( ±3,0 )
the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±c ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2  Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we have:

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>c</mi><mo>=</mo><mo>±</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 2 − b 2    =± 25−9    =± 16    =±4

Therefore, the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±4 ).

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See [link].

<figure class="medium" id="Figure_10_01_007"></figure>

Graph the ellipse given by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 36 + y 2 4 =1. Identify and label the center, vertices, co-vertices, and foci.

center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±6,0 ); co-vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±2 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±4 2 ,0 )

Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form

Graph the ellipse given by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +25 y 2 =100. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

First, use algebra to rewrite the equation in standard form.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo> </mo><mn>4</mn><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 +25 y 2 =100    4 x 2 100 + 25 y 2 100 = 100 100          x 2 25 + y 2 4 =1

Next, we determine the position of the major axis. Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>25</mn><mo>></mo><mn>4</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the major axis is on the x-axis. Therefore, the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =25 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =4. It follows that:

the center of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 )
the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±a,0 )=( ± 25 ,0 )=( ±5,0 )
the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±b )=( 0,± 4 )=( 0,±2 )
the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±c,0 ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we have:

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>c</mi><mo>=</mo><mo>±</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 2 − b 2    =± 25−4    =± 21

Therefore the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ± 21 ,0 ).

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

<figure class="medium" id="Figure_10_01_009"></figure>

Graph the ellipse given by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>49</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +16 y 2 =784. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Standard form:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 16 + y 2 49 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,±7 ); co-vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±4,0 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,± 33 )

Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k ),we use the standard forms<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1, a>b for horizontal ellipses and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1, a>b for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

Given the standard form of an equation for an ellipse centered at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h, k ), sketch the graph.

  1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
    1. If the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then
      • the center is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k )
      • the major axis is parallel to the x-axis
      • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±a,k )
      • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±b )
      • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±c,k )
    2. If the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then
      • the center is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k )
      • the major axis is parallel to the y-axis
      • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±a )
      • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±b,k )
      • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±c )
  2. Solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>using the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 .
  3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an Ellipse Centered at (h, k)

Graph the ellipse given by the equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x+2 ) 2 4 + ( y−5 ) 2 9 =1. Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>9</mn><mo>></mo><mn>4</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the major axis is parallel to the y-axis. Therefore, the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =4 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =9. It follows that:

  • the center of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k )=( −2,5 )
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>±</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>±</mo><msqrt/></mrow></annotation-xml></semantics></math> 9 )=(−2,5±3), or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,2 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,8 )
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>±</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>±</mo><msqrt/></mrow></annotation-xml></semantics></math> 4 ,5)=(−2±2,5), or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −4,5 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,5 )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±c ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>,</mo></mrow></annotation-xml></semantics></math>we have:
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mo>±</mo><msqrt/></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 2 − b 2    =± 9−4    =± 5

Therefore, the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,5− 5 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,5+ 5 ).

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

<figure class="small" id="Figure_10_01_011"></figure>

Graph the ellipse given by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−4 ) 2 36 + ( y−2 ) 2 20 =1. Identify and label the center, vertices, co-vertices, and foci.

Center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,2 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,2 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 10,2 ); co-vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,2−2 5 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,2+2 5 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,2 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 8,2 )

Given the general form of an equation for an ellipse centered at (h, k), express the equation in standard form.

  1. Recognize that an ellipse described by an equation in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><msup/></annotation-xml></semantics></math> x 2 +b y 2 +cx+dy+e=0 is in general form.
  2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
  3. Factor out the coefficients of the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> x 2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> y 2  terms in preparation for completing the square.
  4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> m 1 ( x−h ) 2 + m 2 ( y−k ) 2 = m 3 , where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> m 1 , m 2 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> m 3  are constants.
  5. Divide both sides of the equation by the constant term to express the equation in standard form.
Graphing an Ellipse Centered at (h, k) by First Writing It in Standard Form

Graph the ellipse given by the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9 y 2 −40x+36y+100=0. Identify and label the center, vertices, co-vertices, and foci.

We must begin by rewriting the equation in standard form.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9 y 2 −40x+36y+100=0

Group terms that contain the same variable, and move the constant to the opposite side of the equation.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4 x 2 −40x )+( 9 y 2 +36y )=−100

Factor out the coefficients of the squared terms.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 −10x )+9( y 2 +4y )=−100

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 −10x+25 )+9( y 2 +4y+4 )=−100+100+36

Rewrite as perfect squares.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> ( x−5 ) 2 +9 ( y+2 ) 2 =36

Divide both sides by the constant term to place the equation in standard form.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−5 ) 2 9 + ( y+2 ) 2 4 =1

Now that the equation is in standard form, we can determine the position of the major axis. Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>9</mn><mo>></mo><mn>4</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> the major axis is parallel to the x-axis. Therefore, the equation is in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> ( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =9 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =4. It follows that:

  • the center of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k )=( 5,−2 )
  • the coordinates of the vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±a,k )=( 5± 9 ,−2 )=( 5±3,−2 ), or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−2 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 8,−2 )
  • the coordinates of the co-vertices are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h,k±b )=( 5,−2± 4 )=( 5,−2±2 ), or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,−4 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,0 )
  • the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> h±c,k ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we have:
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>c</mi><mo>=</mo><mo>±</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 2 − b 2    =± 9−4    =± 5

Therefore, the coordinates of the foci are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5− 5 ,−2 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5+ 5 ,−2 ).

Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in[link].

<figure class="medium" id="Figure_10_01_018"></figure>

Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 + y 2 −24x+2y+21=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> (x−3) 2 4 + ( y+1 ) 2 16 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−1 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−5 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,3 ); co-vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,−1 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,−1 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−1−2 3 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−1+2 3 )

Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See [link]. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.

<figure class="small" id="Figure_10_01_013"> <figcaption>Sound waves are reflected between foci in an elliptical room, called a whispering chamber.</figcaption> </figure>
Locating the Foci of a Whispering Chamber

The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in [link].

  1. What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ).
  2. If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.
<figure class="small" id="Figure_10_01_015"></figure>
  1. We are assuming a horizontal ellipse with center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ), so we need to find an equation of the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 a 2 + y 2 b 2 =1, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>We know that the length of the major axis,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>is longer than the length of the minor axis,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.
    • Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we have<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>a</mi><mo>=</mo><mn>96</mn><mo>,</mo></mrow></annotation-xml></semantics></math> so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>=</mo><mn>48</mn><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> a 2 =2304.
    • Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we have<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>b</mi><mo>=</mo><mn>46</mn><mo>,</mo></mrow></annotation-xml></semantics></math> so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>=</mo><mn>23</mn><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> b 2 =529.

    Therefore, the equation of the ellipse is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 2304 + y 2 529 =1.

  2. To find the distance between the senators, we must find the distance between the foci,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±c,0 ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> c 2 = a 2 − b 2 . Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>,</mo></mrow></annotation-xml></semantics></math>we have:
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>c</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 = a 2 − b 2 c 2 =2304−529 Substitute using the values found in part (a).    c=± 2304−529Take the square root of both sides.    c=± 1775   Subtract.    c≈±42 Round to the nearest foot.

    The points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> ±42,0 ) represent the foci. Thus, the distance between the senators is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 42 )=84 feet.

Suppose a whispering chamber is 480 feet long and 320 feet wide.

  1. What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ).
  2. If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 57,600 + y 2 25,600 =1
  2. The people are standing 358 feet apart.

Access these online resources for additional instruction and practice with ellipses.

Key Equations

Horizontal ellipse, center at origin <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 a 2 + y 2 b 2 =1, a>b
Vertical ellipse, center at origin <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 b 2 + y 2 a 2 =1, a>b
Horizontal ellipse, center <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−h ) 2 a 2 + ( y−k ) 2 b 2 =1, a>b
Vertical ellipse, center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−h ) 2 b 2 + ( y−k ) 2 a 2 =1, a>b

Key Concepts

  • An ellipse is the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
  • When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See [link] and [link].
  • When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See [link] and [link].
  • When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See [link] and [link].
  • Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See [link].

Section Exercises

Verbal

Define an ellipse in terms of its foci.

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

Where must the foci of an ellipse lie?

What special case of the ellipse do we have when the major and minor axis are of the same length?

This special case would be a circle.

For the special case mentioned above, what would be true about the foci of that ellipse?

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?

It is symmetric about the x-axis, y-axis, and the origin.

Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +y=4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9 y 2 =36

yes;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 3 2 + y 2 2 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 − y 2 =4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9 y 2 =1

yes;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −8x+9 y 2 −72y+112=0

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 4 + y 2 49 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 2 2 + y 2 7 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,7 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,−7 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,0 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,0 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,3 5 ),( 0,−3 5 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 100 + y 2 64 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 +9 y 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,0 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,0 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 1 3 ),( 0,− 1 3 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 2 3 ,0 ),( − 2 2 3 ,0 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +16 y 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−2 ) 2 49 + ( y−4 ) 2 25 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−2 ) 2 7 2 + ( y−4 ) 2 5 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9,4 ),( −5,4 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,9 ),( 2,−1 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2+2 6 ,4 ),( 2−2 6 ,4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−2 ) 2 81 + ( y+1 ) 2 16 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+5 ) 2 4 + ( y−7 ) 2 9 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+5 ) 2 2 2 + ( y−7 ) 2 3 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5,10 ),( −5,4 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,7 ),( −7,7 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5,7+ 5 ),( −5,7− 5 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−7 ) 2 49 + ( y−7 ) 2 49 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −8x+9 y 2 −72y+112=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−1 ) 2 3 2 + ( y−4 ) 2 2 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,4 ),( −2,4 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,6 ),( 1,2 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1+ 5 ,4 ),( 1− 5 ,4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −54x+9 y 2 −54y+81=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −24x+36 y 2 −360y+864=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−3 ) 2 ( 3 2 ) 2 + ( y−5 ) 2 ( 2 ) 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3+3 2 ,5 ),( 3−3 2 ,5 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,5+ 2 ),( 3,5− 2 ).  Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,5 ),( −1,5 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +24x+16 y 2 −128y+228=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +40x+25 y 2 −100y+100=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+5 ) 2 ( 5 ) 2 + ( y−2 ) 2 ( 2 ) 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,2 ),( −10,2 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5,4 ),( −5,0 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5+ 21 ,2 ),( −5− 21 ,2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 +2x+100 y 2 −1000y+2401=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +24x+25 y 2 +200y+336=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; Endpoints of major axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−4 ),( −8,−4 ). Endpoints of minor axis<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,−2 ),( −3,−6 ). Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3+ 21 ,−4 ),( −3− 21 ,−4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +72x+16 y 2 +16y+4=0

For the following exercises, find the foci for the given ellipses.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+3 ) 2 25 + ( y+1 ) 2 36 =1

Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,−1+ 11 ),( −3,−1− 11 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+1 ) 2 100 + ( y−2 ) 2 4 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =1

Focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 +4 y 2 +4x+8y=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><msup/></mrow></annotation-xml></semantics></math> x 2 + y 2 +200x=0

Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −10,30 ),( −10,−30 )

Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 25 + y 2 36 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 16 + y 2 9 =1

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,0 ),( −4,0 ),(0,3),(0,−3), Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7 ,0 ),( − 7 ,0 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9 y 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>81</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +49 y 2 =1

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 9 ,0 ),( − 1 9 ,0 ),( 0, 1 7 ),( 0,− 1 7 ),  Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 4 2 63 ),( 0,− 4 2 63 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−2 ) 2 64 + ( y−4 ) 2 16 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+3 ) 2 9 + ( y−3 ) 2 9 =1

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,3 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,3 ),( −6,3 ),( −3,0 ),( −3,6 ), Focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,3 ) 

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 2 + ( y+1 ) 2 5 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −8x+16 y 2 −32y−44=0

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,1 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,1 ),( −3,1 ),( 1,3 ),( 1,−1 ), Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,1+4 3 ),( 1,1−4 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 −8x+25 y 2 −100y+91=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 +8x+4 y 2 −40y+112=0

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −4,5 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,5 ),( −6,4 ),( −4,6 ),( −4,4 ), Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −4+ 3 ,5 ),( −4− 3 ,5 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>64</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +128x+9 y 2 −72y−368=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>16</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +64x+4 y 2 −8y+4=0

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,1 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,1 ),( −4,1 ),( −2,5 ),( −2,−3 ), Foci<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,1+2 3 ),( −2,1−2 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>100</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +1000x+ y 2 −10y+2425=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +16x+4 y 2 +16y+16=0

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−2 ), Vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,−2 ),( −4,−2 ),( −2,0 ),( −2,−4 ), Focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−2 )

For the following exercises, use the given information about the graph of each ellipse to determine its equation.

Center at the origin, symmetric with respect to the x- and y-axes, focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math> and point on graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

Center at the origin, symmetric with respect to the x- and y-axes, focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>−2</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math> and point on graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 25 + y 2 29 =1

Center at the origin, symmetric with respect to the x- and y-axes, focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math> and major axis is twice as long as minor axis.

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,2 ); vertex<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9,2 ); one focus:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4+2 6 ,2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−4 ) 2 25 + ( y−2 ) 2 1 =1

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,5 ); vertex<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,11 ); one focus:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3, 5+4 2 )

Center<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,4 ); vertex<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,4 ); one focus:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3+2 3 ,4 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+3 ) 2 16 + ( y−4 ) 2 4 =1

For the following exercises, given the graph of the ellipse, determine its equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 81 + y 2 9 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+2 ) 2 4 + ( y−2 ) 2 9 =1

Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>Area</mtext><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>π</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−3 ) 2 9 + ( y−3 ) 2 16 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>Area = 12π</mtext><mtext> </mtext><mtext>square</mtext><mtext> </mtext><mtext>units</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+6 ) 2 16 + ( y−6 ) 2 36 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x+1 ) 2 4 + ( y−2 ) 2 5 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>Area = 2</mtext><msqrt/></mrow></annotation-xml></semantics></math> 5 π square units

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −8x+9 y 2 −72y+112=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −54x+9 y 2 −54y+81=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>Area = 9π</mtext><mtext> </mtext><mtext>square</mtext><mtext> </mtext><mtext>units</mtext></mrow></annotation-xml></semantics></math>

Real-World Applications

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the height.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 4 h 2 + y 2 1 4 h 2 =1

An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 400 + y 2 144 =1. Distance = 17.32 feet

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Approximately 51.96 feet

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?

Footnotes

  1. 1 Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014.

Glossary

center of an ellipse
the midpoint of both the major and minor axes
conic section
any shape resulting from the intersection of a right circular cone with a plane
ellipse
the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) in a plane such that the sum of their distances from two fixed points is a constant
foci
plural of focus
focus (of an ellipse)
one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) on the ellipse is a constant
major axis
the longer of the two axes of an ellipse
minor axis
the shorter of the two axes of an ellipse