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Mathematics LibreTexts

10-4. Rotation of Axes

Rotation of Axes
In this section, you will:
  • Identify nondegenerate conic sections given their general form equations.
  • Use rotation of axes formulas.
  • Write equations of rotated conics in standard form.
  • Identify conics without rotating axes.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See [link].

<figure id="Figure_10_04_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 nondegenerate conic sections</figcaption> </figure>

Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to thedegenerate conic sections, which are shown in [link]. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.

<figure id="Figure_10_04_002" 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>Degenerate conic sections</figcaption> </figure>

Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0

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></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>are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.

You may notice that the general form equation has an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term that we have not seen in any of the standard form equations. As we will discuss later, the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term rotates the conic whenever<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is not equal to zero.

Conic Sections Example
ellipse <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
circle <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 +4 y 2 =1
hyperbola <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
parabola <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 =9y or 4 y 2 =9x
one line <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>
intersecting lines <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−4 )( y+4 )=0
parallel lines <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−4 )( x−9 )=0
a point <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 +4 y 2 =0
no graph <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 +4 y 2 = − 1
General Form of Conic Sections

A nondegenerate conic section has the general form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0

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></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>are not all zero.

[link] summarizes the different conic sections where<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>0</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><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>are nonzero real numbers. This indicates that the conic has not been rotated.

ellipse <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +C y 2 +Dx+Ey+F=0, A≠C and AC>0
circle <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +C y 2 +Dx+Ey+F=0, A=C
hyperbola <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 −C y 2 +Dx+Ey+F=0 or −A x 2 +C y 2 +Dx+Ey+F=0,where<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>are positive
parabola <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Dx+Ey+F=0 or C y 2 +Dx+Ey+F=0

Given the equation of a conic, identify the type of conic.

  1. Rewrite the equation in the general form, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0.
  2. Identify the values of<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 general form.
    1. If<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>are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse.
    2. If<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>are equal and nonzero and have the same sign, then the graph is a circle.
    3. If<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>are nonzero and have opposite signs, then the graph is a hyperbola.
    4. If either<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>or<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>is zero, then the graph is a parabola.
Identifying a Conic from Its General Form

Identify the graph of each of the following nondegenerate conic sections.

  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 −9 y 2 +36x+36y−125=0
  2. <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> y 2 +16x+36y−10=0
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +3 y 2 −2x−6y−4=0
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>25</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −4 y 2 +100x+16y+20=0
  1. Rewriting the general form, we have

    <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mn>4</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><mi>C</mi><mo>=</mo><mn>−9</mn><mo>,</mo></mrow></annotation-xml></semantics></math> so we observe that<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>have opposite signs. The graph of this equation is a hyperbola.

  2. Rewriting the general form, we have

    <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mn>0</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><mi>C</mi><mo>=</mo><mn>9.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>We can determine that the equation is a parabola, since<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>is zero.

  3. Rewriting the general form, we have

    <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mn>3</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><mi>C</mi><mo>=</mo><mn>3.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>Because<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><mo>,</mo></mrow></annotation-xml></semantics></math> the graph of this equation is a circle.

  4. Rewriting the general form, we have

    <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mn>−25</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><mi>C</mi><mo>=</mo><mn>−4.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>C</mi><mo>></mo><mn>0</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><mi>A</mi><mo>≠</mo><mi>C</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the graph of this equation is an ellipse.

Identify the graph of each of the following nondegenerate conic sections.

  1. <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> y 2 − x 2 +x−4y−9=0
  2. <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 +4 y 2 +16x+49y−81=0
  1. hyperbola
  2. ellipse

Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term, which aligns the graphs with the x- and y-axes. When we add an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>,</mo></mrow></annotation-xml></semantics></math>then every point on the plane may be thought of as having two representations:<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 Cartesian plane with the originalx-axis and y-axis, 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> x ′ , y ′ ) on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See [link].

<figure class="small" id="Figure_10_04_003"> <figcaption>The graph of the rotated ellipse<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 + y 2 –xy–15=0</figcaption> </figure>

We will find the relationships between<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on the Cartesian plane with<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 ′  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 ′  on the new rotated plane. See [link].

<figure class="small" id="Figure_10_04_004"> <figcaption>The Cartesian plane with x- and y-axes and the resulting x′− and y′−axes formed by a rotation by an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math></figcaption> </figure>

The original coordinate x- and y-axes have unit vectors<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</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>j</mi><mtext>  </mtext></mrow><mo>.</mo></annotation-xml></semantics></math>The rotated coordinate axes have unit vectors<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> i ′  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> j ′ .The angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is known as the angle of rotation. See [link]. We may write the new unit vectors in terms of the original ones.

<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>i</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ =cos θi+sin θj j ′ =−sin θi+cos θj
<figure class="small" id="Figure_10_04_005"> <figcaption>Relationship between the old and new coordinate planes.</figcaption> </figure>

Consider a vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the new coordinate plane. It may be represented in terms of its coordinate axes.

<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>u</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ i ′ + y ′ j ′ u= x ′ (i cos θ+j sin θ)+ y ′ (−i sin θ+j cos θ) Substitute. u=ix' cos θ+jx' sin θ−iy' sin θ+jy' cos θ Distribute.u=ix' cos θ−iy' sin θ+jx' sin θ+jy' cos θ Apply commutative property. u=(x' cos θ−y' sin θ)i+(x' sin θ+y' cos θ)jFactor by grouping.

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ i ′ + y ′ j ′ , we have representations of<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of the new coordinate system.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mi>x</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ and y= x ′ sin θ+ y ′ cos θ
Equations of Rotation

If 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> x,y ) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>from the positive x-axis, then the coordinates of the point with respect to the new axes 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> x ′ , y ′ ). We can use the following equations of rotation to define the relationship between<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 ) 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> x ′ , y ′ ):

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ

and

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ

Given the equation of a conic, find a new representation after rotating through an angle.

  1. Find<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ.
  2. Substitute the expression for<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>into in the given equation, then simplify.
  3. Write the equations with<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 ′  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 ′  in standard form.
Finding a New Representation of an Equation after Rotating through a Given Angle

Find a new representation of the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −xy+2 y 2 −30=0 after rotating through an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>45°</mn><mo>.</mo></mrow></annotation-xml></semantics></math>

Find<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>,</mo></mrow></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ.

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>45°</mn><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 columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ cos( 45° )− y ′ sin( 45° ) x= x ′ ( 1 2 )− y ′ ( 1 2 ) x= x ′ − y ′ 2

and

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math> x ′ sin(45°)+ y ′ cos(45°) y= x ′ ( 1 2 )+ y ′ ( 1 2 ) y= x ′ + y ′ 2

Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cosθ− y ′ sinθ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ into<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −xy+2 y 2 −30=0.

<math display="block" 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 ′ − y ′ 2 ) 2 −( x ′ − y ′ 2 )( x ′ + y ′ 2 )+2 ( x ′ + y ′ 2 ) 2 −30=0

Simplify.

<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><menclose notation="updiagonalstrike"><mn>2</mn></menclose></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( x ′ − y ′ )( x ′ − y ′ ) 2 − ( x ′ − y ′ )( x ′ + y ′ ) 2 + 2 ( x ′ + y ′ )( x ′ + y ′ ) 2 −30=0 FOIL method             x ′ 2 −2 x ′ y ′ + y′ 2 − ( x ′ 2 − y ′ 2 ) 2 + x ′ 2 +2 x ′ y ′ + y ′ 2 −30=0 Combine like terms.                                                              2 x ′ 2 +2 y ′2 − ( x ′ 2 − y ′ 2 ) 2 =30 Combine like terms.                                                        2( 2 x ′ 2 +2 y ′ 2 − ( x ′ 2 − y ′ 2 ) 2 )=2(30)Multiply both sides by 2.                                                               4 x ′ 2 +4 y ′ 2 −( x ′ 2 − y ′ 2 )=60 Simplify.                                                                 4 x ′ 2 +4 y ′ 2 − x ′ 2 + y ′ 2 =60 Distribute.                                                                                     3 x ′ 2 60 + 5 y ′ 2 60 = 60 60 Set equal to 1.

Write the equations with<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 ′  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 ′  in the standard form.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><msup><mi>x</mi></msup></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> ′ 2 20 + y ′ 2 12 =1

This equation is an ellipse. [link] shows the graph.

<figure class="small" id="Figure_10_04_006"></figure>

Writing Equations of Rotated Conics in Standard Form

Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0 into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in 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 ′  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 ′  coordinate system without 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 ′ y ′  term, by rotating the axes by a measure of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>that satisfies

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= A−C B

We have learned already that any conic may be represented by the second degree equation

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0

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></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>are not all zero. However, 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><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> then we have an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= A−C B .

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is in the first quadrant, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is between<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>45°</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>
If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is in the second quadrant, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is between<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>45°</mn><mo>,</mo><mn>90°</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>
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>C</mi><mo>,</mo></mrow></annotation-xml></semantics></math> then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>45°</mn><mo>.</mo></mrow></annotation-xml></semantics></math>

 

Given an equation for a conic in 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 ′ y ′  system, rewrite the equation without 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 ′ y ′  term in terms of<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 ′  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 ′ ,where 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 ′  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 ′  axes are rotations of the standard axes by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>degrees.

  1. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>
  2. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</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>cos</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  3. Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</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>cos</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>into<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ.
  4. Substitute the expression for<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>into in the given equation, and then simplify.
  5. Write the equations with<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 ′  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 ′  in the standard form with respect to the rotated axes.
Rewriting an Equation with respect to the x′ and y′ axes without the x′y′ Term

Rewrite the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>8</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −12xy+17 y 2 =20 in 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 ′ y ′  system without an<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 ′ y ′  term.

First, we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>See [link].

<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><mn>8</mn><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 −12xy+17 y 2 =20⇒A=8, B=−12 and C=17                     cot(2θ)= A−C B = 8−17 −12                     cot(2θ)= −9 −12 = 34
<figure class="small" id="Figure_10_04_007"></figure>
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= 3 4 = adjacent opposite

So the hypotenuse is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><msup><mn>3</mn></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 + 4 2 = h 2 9+16= h 2 25= h 2 h=5   

Next, we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</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 columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1−cos(2θ) 2 = 1− 3 5 2 = 5 5 − 3 5 2 = 5−3 5 ⋅ 1 2 = 2 10 = 1 5 sin θ= 1 5 cos θ= 1+cos(2θ) 2 = 1+ 3 5 2 = 5 5 + 3 5 2= 5+3 5 ⋅ 1 2 = 8 10 = 4 5 cos θ= 2 5

Substitute the values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</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>cos</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>into<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ.

<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/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ x= x ′ ( 2 5 )− y ′ ( 1 5 ) x= 2 x ′ − y ′ 5

and

<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><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ y= x ′ ( 1 5 )+ y ′ ( 2 5 ) y= x ′ +2 y ′ 5

Substitute the expressions for<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>into in the given equation, and then simplify.

<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><mn>8</mn><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( 2 x ′ − y ′ 5 ) 2 −12( 2 x ′ − y ′ 5 )( x ′ +2 y ′ 5 )+17 ( x ′ +2 y ′ 5 ) 2 =20      8( (2 x ′ − y ′ )(2 x ′ − y ′) 5 )−12( (2 x ′ − y ′ )( x ′ +2 y ′ ) 5 )+17( ( x ′ +2 y ′ )( x ′ +2 y ′ ) 5 )=20        8( 4 x ′ 2 −4 x ′ y ′ + y ′ 2 )−12( 2 x ′ 2 +3 x ′ y ′−2 y ′ 2 )+17( x ′ 2 +4 x ′ y ′ +4 y ′ 2 )=100 32 x ′ 2 −32 x ′ y ′ +8 y ′ 2 −24 x ′ 2 −36 x ′ y ′ +24 y ′ 2 +17 x ′ 2 +68 x ′ y ′ +68y ′ 2 =100                                                                                                   25 x ′ 2 +100 y ′ 2 =100                                                                                                  25 100 x ′ 2 + 100 100 y ′ 2 = 100 100  

Write the equations with<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 ′  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 ′  in the standard form with respect to the new coordinate system.

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

[link] shows the graph of the ellipse.

<figure class="small" id="Figure_10_04_008"></figure>

Rewrite the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>13</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −6 3 xy+7 y 2 =16 in 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 ′ y ′  system without 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 ′ y ′  term.

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

Graphing an Equation That Has No x′y′ Terms

Graph the following equation relative to 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 ′ y ′  system:

<math display="block" 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 +12xy−4 y 2 =30

First, we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ ).

<math display="block" 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 +12xy−4 y 2 =20⇒A=1, B=12,and C=−4
<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>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> A−C B cot(2θ)= 1−(−4) 12 cot(2θ)= 5 12

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= 5 12 , we can draw a reference triangle as in [link].

<figure class="small" id="Figure_10_04_009"></figure>
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= 5 12 = adjacent opposite

Thus, the hypotenuse is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><msup><mn>5</mn></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 + 12 2 = h 2 25+144= h 2 169= h 2 h=13

Next, we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</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>cos</mi><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>We will use half-angle identities.

<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><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1−cos(2θ) 2 = 1− 5 13 2 = 13 13 − 5 13 2 = 8 13 ⋅ 1 2 = 2 13 cos θ= 1+cos(2θ) 2 = 1+ 5 13 2 = 13 13 + 5 13 2 = 18 13⋅ 1 2 = 3 13

Now we find<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext>. </mtext></mrow></annotation-xml></semantics></math>

<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/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ x= x ′ ( 3 13 )− y ′ ( 2 13 ) x= 3 x ′ −2 y ′ 13

and

<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/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ sin θ+ y ′ cos θ y= x ′ ( 2 13 )+ y ′ ( 3 13 ) y= 2 x ′ +3 y ′ 13

Now we substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 x ′ −2 y ′ 13  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 x ′ +3 y ′ 13  into<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 +12xy−4 y 2 =30.

<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><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( 3 x ′ −2 y ′ 13 ) 2 +12( 3 x ′ −2 y ′ 13 )( 2 x ′ +3 y ′ 13 )−4 ( 2 x ′ +3 y ′ 13 ) 2 =30                                    ( 1 13 )[ (3 x ′ −2 y ′ ) 2 +12(3 x ′ −2 y ′ )(2 x ′ +3 y ′ )−4 (2 x ′ +3 y ′ ) 2 ]=30  Factor. ( 1 13 )[ 9 x ′ 2−12 x ′ y ′ +4 y ′ 2 +12( 6 x ′ 2 +5 x ′ y ′ −6 y ′ 2 )−4( 4 x ′ 2 +12 x ′ y ′ +9 y ′ 2 ) ]=30 Multiply.   ( 1 13 )[ 9 x ′ 2 −12 x ′ y ′+4 y ′ 2 +72 x ′ 2 +60 x ′ y ′ −72 y ′ 2 −16 x ′ 2 −48 x ′ y ′ −36 y ′ 2 ]=30 Distribute.                                                                                                  ( 1 13 )[ 65 x ′ 2 −104 y ′ 2 ]=30 Combine like terms.                                                                                                           65 x ′ 2 −104 y ′ 2 =390 Multiply.                                                                                                                                   x ′ 2 6 − 4 y ′ 2 15 =1  Divide by 390.

[link] shows the graph of the hyperbola<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 6 − 4 y ′ 2 15 =1.      

<figure class="small" id="Figure_10_04_010"></figure>

Identifying Conics without Rotating Axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0

If we apply the rotation formulas to this equation we get the form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>A</mi></msup></mrow></annotation-xml></semantics></math> ′ x ′ 2 + B ′ x ′ y ′ + C ′ y ′ 2 + D ′ x ′ + E ′ y ′ + F ′ =0

It may be shown that<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 −4AC= B ′ 2 −4 A ′ C ′ . The expression does not vary after rotation, so we call the expression invariant. The discriminant,<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 −4AC, is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

Using the Discriminant to Identify a Conic

If the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0 is transformed by rotating axes into 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> A ′ x ′ 2 + B ′ x ′ y ′ + C ′ y ′ 2 + D ′ x ′ + E ′ y ′ + F ′ =0, then<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 −4AC= B ′ 2 −4 A ′ C ′ .

The equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0 is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.

If the discriminant,<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 −4AC,is

  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic section is an ellipse
  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic section is a parabola
  • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic section is a hyperbola
Identifying the Conic without Rotating Axes

Identify the conic for each of the following without rotating axes.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+2 y 2 −5=0
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+12 y 2 −5=0
  1. Let’s begin by determining<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> and<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><munder><mrow><munder><mn>5</mn></munder></mrow></munder></mrow></annotation-xml></semantics></math> ︸ A x 2 + 2 3 ︸ B xy+ 2 ︸ C y 2 −5=0

    Now, we find the discriminant.

    <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>B</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 −4AC= ( 2 3 ) 2 −4(5)(2)                 =4(3)−40                 =12−40                 =−28<0

    Therefore,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+2 y 2 −5=0 represents an ellipse.

  2. Again, let’s begin by determining<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> and<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><munder><munder><mn>5</mn></munder></munder></mrow></annotation-xml></semantics></math> ︸ A x 2 + 2 3 ︸ B xy+ 12 ︸ C y 2 −5=0

    Now, we find the discriminant.

    <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>B</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 −4AC= ( 2 3 ) 2 −4(5)(12)                 =4(3)−240                 =12−240                 =−228<0

    Therefore,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+12 y 2 −5=0 represents an ellipse.

Identify the conic for each of the following without rotating axes.

  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 −9xy+3 y 2 −12=0
  2. <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 −9xy+4 y 2 −4=0
  1. hyperbola
  2. ellipse

Access this online resource for additional instruction and practice with conic sections and rotation of axes.

Key Equations

General Form equation of a conic section <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0
Rotation of a conic section <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>x</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x ′ cos θ− y ′ sin θ y= x ′ sin θ+ y ′ cos θ
Angle of rotation <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>,</mo><mtext>where </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= A−C B

Key Concepts

  • Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
  • A nondegenerate conic section has the general form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0 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><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>are not all zero. The values of<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> 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>determine the type of conic. See [link].
  • Equations of conic sections with an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term have been rotated about the origin. See [link].
  • The general form can be transformed into an equation in 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 ′  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 ′  coordinate system without 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 ′ y ′  term. See [link] and [link].
  • An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See [link].

Section Exercises

Verbal

What effect does the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term have on the graph of a conic section?

The<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term causes a rotation of the graph to occur.

If the equation of a conic section is written in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +B y 2 +Cx+Dy+E=0 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> what can we conclude?

If the equation of a conic section is written in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0,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 −4AC>0, what can we conclude?

The conic section is a hyperbola.

Given the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +4x+3 y 2 −12=0, what can we conclude 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><mn>0</mn><mo>?</mo></mrow></annotation-xml></semantics></math>

For the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +Bxy+C y 2 +Dx+Ey+F=0, the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>that satisfies<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )= A−C B  gives us what information?

It gives the angle of rotation of the axes in order to eliminate the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term.

Algebraic

For the following exercises, determine which conic section is represented based on the given equation.

<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 +4 y 2 +72x+36y−500=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 −10x+4y−10=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> parabola

<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 −2 y 2 +4x−6y−2=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 − y 2 +8x−1=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mo>−</mo><mn>4</mn><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> hyperbola

<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> y 2 −5x+9y+1=0

<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 +3 y 2 −8x−12y+2=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mn>6</mn><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> ellipse

<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 +9xy+4 y 2 −36y−125=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +6xy+3 y 2 −36y−125=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2 −4AC=0, parabola

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +3 3 xy−4 y 2 +9=0

<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 +4 3 xy+6 y 2 −6x−3=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2 −4AC=0, parabola

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> x 2 +4 2 xy+2 y 2 −2y+1=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +4 2 xy+4 y 2 −10x+1=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2 −4AC=−96<0, ellipse

For the following exercises, find a new representation of the given equation after rotating through the given angle.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +xy+3 y 2 −5=0,θ=45°

<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 −xy+4 y 2 −2=0,θ=45°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>7</mn><msup/></mrow></annotation-xml></semantics></math> x ′ 2 +9 y ′ 2 −4=0

<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 +8xy−1=0,θ=30°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +8xy+1=0,θ=45°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x ′ 2 +2 x ′ y ′ −5 y ′ 2 +1=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 + 2 xy+4 y 2 +y+2=0,θ=45°

For the following exercises, determine the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>that will eliminate the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term and write the corresponding equation without the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term.

<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 +3 3 xy+4 y 2 +y−2=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> 60 ∘ ,11 x ′ 2 − y ′ 2 + 3 x ′ + y ′ −4=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 +2 3 xy+6 y 2 +y−2=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 −3 3 xy+6 y 2 +4y−3=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> 150 ∘ ,21 x ′ 2 +9 y ′ 2 +4 x ′ −4 3 y ′ −6=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 − 3 xy−2 y 2 −x=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 +24xy+9 y 2 +6x−6y+2=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>≈</mo><msup/></mrow></annotation-xml></semantics></math> 36.9 ∘ ,125 x ′ 2 +6 x ′ −42 y ′ +10=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 +4xy+4 y 2 +3x−2=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 +4xy+ y 2 −2x+1=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> 45 ∘ ,3 x ′ 2 − y ′ 2 − 2 x ′ + 2 y ′ +1=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 −2 3 xy+6 y 2 −1=0

Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> x 2 ,θ=− 45 ∘

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> y 2 ,θ= 45 ∘

<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 1 =1,θ= 45 ∘

<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 ′ − y ′ ) 2 8 + ( x ′ + y ′ ) 2 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>y</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 16 + x 2 9 =1,θ= 45 ∘

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>y</mi></msup></mrow></annotation-xml></semantics></math> 2 − x 2 =1,θ= 45 ∘

<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 ′ + y ′ ) 2 2 − ( x ′ − y ′ ) 2 2 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 2 ,θ= 30 ∘

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( y−1 ) 2 ,θ= 30 ∘

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

<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 9 + y 2 4 =1,θ= 30 ∘

For the following exercises, graph the equation relative to 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 ′ y ′  system in which the equation has no<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 ′ y ′  term.

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

<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 +10xy+ y 2 −6=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 −10xy+ y 2 −24=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 −3 3 xy+ y 2 −22=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+4 y 2 −21=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>11</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +10 3 xy+ y 2 −64=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>21</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +2 3 xy+19 y 2 −18=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 +24xy+9 y 2 −130x+90y=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 +24xy+9 y 2 −60x+80y=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>13</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −6 3 xy+7 y 2 −16=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 −4xy+ y 2 −8 5 x−16 5 y=0

For the following exercises, determine the angle of rotation in order to eliminate the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>term. Then graph the new set of axes.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −5 3 xy+ y 2 +10x−12y=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −5xy+6 y 2 +20x−y=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> 45 ∘

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −8 3 xy+14 y 2 +10x−3y=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 +6 3 xy+10 y 2 +20x−40y=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> 60 ∘

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +3xy+4 y 2 +2x−4=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 +24xy+9 y 2 +20x−44y=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>≈</mo><msup/></mrow></annotation-xml></semantics></math> 36.9 ∘

For the following exercises, determine the value of<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>based on the given equation.

Given<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 +kxy+16 y 2 +8x+24y−48=0, find<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>for the graph to be a parabola.

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +kxy+12 y 2 +10x−16y+28=0, find<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>for the graph to be an ellipse.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>4</mn><msqrt/></mrow></annotation-xml></semantics></math> 6 <k<4 6

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +kxy+4 y 2 −6x+20y+128=0, find<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>for the graph to be a hyperbola.

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><msup/></mrow></annotation-xml></semantics></math> x 2 +8xy+8 y 2 −12x+16y+18=0, find<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>for the graph to be a parabola.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></annotation-xml></semantics></math>

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>6</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +12xy+k y 2 +16x+10y+4=0, find<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>for the graph to be an ellipse.

Glossary

angle of rotation
an acute angle formed by a set of axes rotated from the Cartesian plane where, if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )>0,then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is between<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>45°</mn><mo stretchy="false">)</mo><mo>;</mo></mrow></annotation-xml></semantics></math>if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math>then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is between<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>45°</mn><mo>,</mo><mn>90°</mn><mo stretchy="false">)</mo><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ )=0,then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>45°</mn></mrow></annotation-xml></semantics></math>
degenerate conic sections
any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.
nondegenerate conic section
a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas