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Mathematics LibreTexts

10-5. Conic Sections in Polar Coordinates

Conic Sections in Polar Coordinates
In this section, you will:
  • Identify a conic in polar form.
  • Graph the polar equations of conics.
  • Define conics in terms of a focus and a directrix.
<figure id="Figure_10_05_008" 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>Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr)</figcaption> </figure>

Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets’ orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.

In an elliptical orbit, the periapsis is the point at which the two objects are closest, and the apoapsis is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body’s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.

Identifying a Conic in Polar Form

Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Consider the parabola<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><mn>2</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> y 2  shown in [link].

<figure class="small" id="Figure_10_05_001"></figure>

In The Parabola, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>at the pole, and a line, the directrix, which is perpendicular to the polar axis.

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>F</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a fixed point, the focus, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a fixed line, the directrix, then we can let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>be a fixed positive number, called theeccentricity, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> PF PD  is a conic. In other words, we can define a conic as the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with the property that the ratio of the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>F</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to the distance from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is equal to the constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

For a conic with eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><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><mn>0</mn><mo>≤</mo><mi>e</mi><mo><</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is an ellipse
  • if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is a parabola
  • if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>></mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is an hyperbola

With this definition, we may now define a conic in terms of the directrix,<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><mo>±</mo><mi>p</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and 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><mo>.</mo></mrow></annotation-xml></semantics></math> Thus, each conic may be written as a polar equation, an equation written in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</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>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

The Polar Equation for a Conic

For a conic with a focus at the origin, if the directrix is<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><mo>±</mo><mi>p</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>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a positive real number, and the eccentricity is a positive real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the conic has a polar equation

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> ep 1±e cos θ

For a conic with a focus at the origin, if the directrix is<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><mo>±</mo><mi>p</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>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is a positive real number, and the eccentricity is a positive real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the conic has a polar equation

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> ep 1±e sin θ

Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.

  1. Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
  2. Identify the eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>as the coefficient of the trigonometric function in the denominator.
  3. Compare<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with 1 to determine the shape of the conic.
  4. Determine the directrix as<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><mi>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>if cosine is in the denominator 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><mi>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>if sine is in the denominator. Set<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mi>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equal to the numerator in standard form to solve 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>or<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>
Identifying a Conic Given the Polar Form

For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 3+2 sin θ
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 4+5 cos θ
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 2−2 sin θ

For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 c , where<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 that constant.

  1. Multiply the numerator and denominator by<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> 1 3 .
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 3+2sin θ ⋅ ( 1 3 ) ( 1 3 ) = 6( 1 3 ) 3( 1 3 )+2( 1 3 )sin θ = 2 1+ 2 3  sin θ

    Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math> is in the denominator, the directrix is<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><mi>p</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Comparing to standard form, note that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3 .Therefore, from the numerator,

    <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>2</mn><mo>=</mo><mi>e</mi><mi>p</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>      2= 2 3 p ( 3 2 )2=( 3 2 ) 2 3 p      3=p

    Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo><</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is an ellipse. The eccentricity is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3and the directrix is<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><mn>3.</mn></mrow></annotation-xml></semantics></math>

  2. Multiply the numerator and denominator by<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> 1 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><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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 12 4+5 cos θ ⋅ ( 1 4 ) ( 1 4 ) r= 12( 1 4 ) 4( 1 4 )+5( 1 4 )cos θ r= 3 1+ 5 4  cos θ

    Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> cos</mtext><mtext> </mtext><mi>θ</mi><mo> </mo></mrow></annotation-xml></semantics></math>is in the denominator, the directrix is<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><mi>p</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Comparing to standard form,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 4 . Therefore, from the numerator,

    <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>3</mn><mo>=</mo><mi>e</mi><mi>p</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>       3= 5 4 p  ( 4 5 )3=( 4 5 ) 5 4 p      12 5 =p

    Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>></mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is a hyperbola. The eccentricity is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 4  and the directrix is<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> 12 5 =2.4.

  3. Multiply the numerator and denominator by<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> 1 2 .
    <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/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 7 2−2 sin θ ⋅ ( 1 2 ) ( 1 2 ) r= 7( 1 2 ) 2( 1 2 )−2( 1 2 ) sin θ r= 7 2 1−sin θ

    Because sine is in the denominator, the directrix is<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><mi>−</mi><mi>p</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Comparing to standard form,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>Therefore, from the numerator,

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mfrac><mn>7</mn></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> 2 =ep 7 2 =( 1 )p 7 2 =p

    Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> the conic is a parabola. The eccentricity is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the directrix is<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><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 2 =−3.5.

Identify the conic with focus at the origin, the directrix, and the eccentricity for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3−cos θ .

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

Graphing the Polar Equations of Conics

When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and, therefore, the shape of the curve. The next step is to substitute values for<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>and solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to plot a few key points. Setting<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>equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo>,</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 ,π, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 3π 2  provides the vertices so we can create a rough sketch of the graph.

Graphing a Parabola in Polar Form

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 3+3 cos θ .

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 3 .

<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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5 3+3 cos θ = 5( 1 3 ) 3( 1 3 )+3( 1 3 )cos θ r= 5 3 1+cos θ

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math>we will graph a parabola with a focus at the origin. The function has a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo> </mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and there is an addition sign in the denominator, so the directrix is<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><mi>p</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"><mtable columnalign="left"><mtr><mtd><mfrac><mn>5</mn></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> 3 =ep 5 3 =(1)p 5 3 =p

The directrix is<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> 5 3 .

Plotting a few key points as in [link] will enable us to see the vertices. See [link].

  A B C D
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>θ</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>π</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 2
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 3+3 cos θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>5</mn></mfrac></mrow></annotation-xml></semantics></math> 6 ≈0.83 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>5</mn></mfrac></mrow></annotation-xml></semantics></math> 3 ≈1.67 undefined <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>5</mn></mfrac></mrow></annotation-xml></semantics></math> 3 ≈1.67
<figure class="small" id="Figure_10_05_002"></figure>
Analysis

We can check our result with a graphing utility. See [link].

<figure class="small" id="Figure_10_05_003"></figure>
Graphing a Hyperbola in Polar Form

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 2−3 sin θ .

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2 .

<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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 8 2−3sin θ = 8( 1 2 ) 2( 1 2 )−3( 1 2 )sin θ r= 4 1− 3 2  sin θ

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 ,e>1, so we will graph a hyperbola with a focus at the origin. The function has a<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>term and there is a subtraction sign in the denominator, so the directrix is<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><mi>−</mi><mi>p</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><mtext>      </mtext><mn>4</mn><mo>=</mo><mi>e</mi><mi>p</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>       4=( 3 2 )p 4( 2 3 )=p       8 3 =p

The directrix is<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><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 3 .

Plotting a few key points as in [link] will enable us to see the vertices. See [link].

  A B C D
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>θ</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>π</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 2
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 2−3sin θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>4</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>8</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>4</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>8</mn></mfrac></mrow></annotation-xml></semantics></math> 5 =1.6
<figure id="Figure_10_05_004"></figure>
Graphing an Ellipse in Polar Form

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 5−4 cos θ .

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 5 .

<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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 10 5−4cos θ = 10( 1 5 ) 5( 1 5 )−4( 1 5 )cos θ r= 2 1− 4 5  cos θ

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5 ,e<1, so we will graph an ellipse with a focus at the origin. The function has a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mtext> </mtext><mi>θ</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and there is a subtraction sign in the denominator, so the directrix is<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><mi>−</mi><mi>p</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><mtext>      </mtext><mn>2</mn><mo>=</mo><mi>e</mi><mi>p</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>       2=( 4 5 )p 2( 5 4 )=p       5 2 =p

The directrix is<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><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 2 .

Plotting a few key points as in [link] will enable us to see the vertices. See [link].

  A B C D
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>θ</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>π</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 2
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 5−4 cos θ <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>2</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>10</mn></mrow></mfrac></mrow></annotation-xml></semantics></math> 9 ≈1.1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>2</mn></annotation-xml></semantics></math>
<figure class="small" id="Figure_10_05_006"></figure>
Analysis

We can check our result using a graphing utility. See [link].

<figure class="small" id="Figure_10_05_007"> <figcaption><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 5−4 cos θ  graphed on a viewing window of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> –3,12,1 ] by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mo>–</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>θ</mi><mtext> </mtext><mtext>min =</mtext><mtext> </mtext><mn>0</mn></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>θ</mi><mtext> </mtext><mtext>max =</mtext><mtext> </mtext><mn>2</mn><mi>π</mi><mo>.</mo></mrow></annotation-xml></semantics></math></figcaption> </figure>

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 4−cos θ .

Defining Conics in Terms of a Focus and a Directrix

So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.

Given the focus, eccentricity, and directrix of a conic, determine the polar equation.

  1. Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of<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> we use the general polar form in terms of sine. If the directrix is given in terms of<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></mrow></annotation-xml></semantics></math> we use the general polar form in terms of cosine.
  2. Determine the sign in the denominator. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>p</mi><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> use subtraction. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>p</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> use addition.
  3. Write the coefficient of the trigonometric function as the given eccentricity.
  4. Write the absolute value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>p</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> in the numerator, and simplify the equation.
Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix

Find the polar form of the conic given a focus at the origin,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and directrix<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><mo>−</mo><mn>2.</mn></mrow></annotation-xml></semantics></math>

The directrix is<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><mi>−</mi><mi>p</mi><mo>,</mo></mrow></annotation-xml></semantics></math> so we know the trigonometric function in the denominator is sine.

Because<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><mn>−2</mn><mo>,</mo><mn>–2</mn><mo><</mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math> so we know there is a subtraction sign in the denominator. We use the standard form of

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> ep 1−e sin θ

and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</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><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> −2 |=2=p.

Therefore,

<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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (3)(2) 1−3 sin θ r= 6 1−3 sin θ
Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix

Find the polar form of a conic given a focus at the origin,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5 , and directrix<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><mn>4.</mn></mrow></annotation-xml></semantics></math>

Because the directrix is<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><mi>p</mi><mo>,</mo></mrow></annotation-xml></semantics></math>we know the function in the denominator is cosine. Because<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><mn>4</mn><mo>,</mo><mn>4</mn><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math>so we know there is an addition sign in the denominator. We use the standard form of

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> ep 1+e cos θ

and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> 4 |=4=p.

Therefore,

<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>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ( 3 5 )(4) 1+ 3 5  cos θ r= 12 5 1+ 3 5  cos θ r= 12 5 1( 5 5 )+ 3 5  cos θ r= 12 5 5 5 + 3 5  cos θ r= 12 5 ⋅ 5 5+3 cos θ r= 125+3 cos θ

Find the polar form of the conic given a focus at the origin,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> and directrix<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><mn>−1.</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 1−cosθ

Converting a Conic in Polar Form to Rectangular Form

Convert the conic<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 5−5sin θto rectangular form.

We will rearrange the formula to use the identities<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo> </mo><mi>r</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 + y 2 ,x=r cos θ,and y=r sin 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><mtext>                          </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 5−5 sin θ  r⋅(5−5 sin θ)= 1 5−5 sin θ ⋅(5−5 sin θ) Eliminate the fraction.         5r−5r sin θ=1 Distribute.                        5r=1+5r sin θ Isolate 5r.                     25 r 2 = (1+5r sin θ) 2 Square both sides.          25( x 2 + y 2 )= (1+5y) 2Substitute r= x 2 + y 2  and y=r sin θ.         25 x 2 +25 y 2 =1+10y+25 y 2 Distribute and use FOIL.           25 x 2 −10y=1Rearrange terms and set equal to 1.

Convert the conic<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1+2 cos θ  to rectangular form.

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

Access these online resources for additional instruction and practice with conics in polar coordinates.

Visit this website for additional practice questions from Learningpod.

Key Concepts

  • Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • A conic is the set of all points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> PF PD , where eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a positive real number. Each conic may be written in terms of its polar equation. See [link].
  • The polar equations of conics can be graphed. See [link], [link], and [link].
  • Conics can be defined in terms of a focus, a directrix, and eccentricity. See [link] and [link].
  • We can use the identities<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 + y 2 ,x=r cos 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><mi>r</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to convert the equation for a conic from polar to rectangular form. See [link].

Section Exercises

Verbal

Explain how eccentricity determines which conic section is given.

If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

If a conic section is written as a polar equation, what must be true of the denominator?

If a conic section is written as a polar equation, and the denominator involves<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><mo>,</mo></mrow></annotation-xml></semantics></math>what conclusion can be drawn about the directrix?

The directrix will be parallel to the polar axis.

If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?

What do we know about the focus/foci of a conic section if it is written as a polar equation?

One of the foci will be located at the origin.

Algebraic

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 1−2 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 4−4 sin θ

Parabola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and directrix<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> 3 4  units below the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 4−3 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 1+2 sin θ

Hyperbola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and directrix<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> 5 2  units above the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 16 4+3 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 10+10 cos θ

Parabola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and directrix<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> 3 10  units to the right of the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1−cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 7+2 cos θ

Ellipse with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 7  and directrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>units to the right of the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>3</mn><mo>+</mo><mn>5</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>11</mn></mrow></annotation-xml></semantics></math>

Hyperbola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 3  and directrix<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> 11 5  units above the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mn>5</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>7</mn><mo>+</mo><mn>8</mn><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></mrow></annotation-xml></semantics></math>

Hyperbola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 7  and directrix<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> 7 8  units to the right of the pole.

For the following exercises, convert the polar equation of a conic section to a rectangular equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 1+3 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 5−3 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>25</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +16 y 2 −12y−4=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 3−2 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2+5 cos θ

<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 −4 y 2 −30x+9=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 2+2 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 8−8 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>64</mn><msup/></mrow></annotation-xml></semantics></math> y 2 =48x+9

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 6+7 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 5−11 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>96</mn><msup/></mrow></annotation-xml></semantics></math> y 2 −25 x 2 +110y+25=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>5</mn><mo>+</mo><mn>2</mn><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<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 +4 y 2 −2x−1=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>2.5</mn><mo>−</mo><mn>2.5</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6sec θ −2+3 sec θ

<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 +9 y 2 −24x−36=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6csc θ 3+2 csc θ

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 2+cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3+3 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 5−4 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 1+2 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 4−5 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 4−4 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1−sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 3+2 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>3</mn><mo>−</mo><mn>4</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><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><mi>r</mi><mo stretchy="false">(</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>6</mn><mo>−</mo><mn>4</mn><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5+cosθ

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

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

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

Directrix: <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><mn>2</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>1</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 1+cosθ

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

Directrix: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 4 ; e= 7 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 8−28cosθ

Directrix:<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> 2 5 ; e= 7 2

Directrix: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 2+3sinθ

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−2</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 3

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−5</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 15 4−3cosθ

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mn>2.5</mn></mrow></annotation-xml></semantics></math>

Directrix:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−3</mn><mo>;</mo><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 3−3cosθ

Extensions

Recall from Rotation of Axes that equations of conics 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 rotated graphs. For the following exercises, express each equation in polar form with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>as a function 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></mrow></annotation-xml></semantics></math>

<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>2</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 +xy+ y 2 =4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>±</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1+sinθcosθ

<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 +4xy+2 y 2 =9

<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 =4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>±</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 4cosθ+3sinθ

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

Chapter Review Exercises

The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 25 + y 2 64 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 5 2 + y 2 8 2 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,0 ),( −5,0 ),( 0,8 ),( 0,−8 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 39 ),( 0,− 39 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 100 + ( y+3 ) 2 36 =1

<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 + y 2 +54x−4y+76=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 1 2 + (y−2) 2 3 2 =1  (−3,2);  (−2,2),(−4,2),(−3,5),(−3,−1);  ( −3,2+2 2 ),( −3,2−2 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> x 2 +36 y 2 −36x+72y+36=0

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 36 + y 2 9 =1

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

 

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 25 + ( y+3 ) 2 49 =1

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

center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−2 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−2+4 3 , ),( −2,−2−4 3 )

 

 

<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 +3 y 2 −20x+12y+38=0

For the following exercises, use the given information to find the equation for the ellipse.

Center at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ),focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,0 ),vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5,0 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 25 + y 2 16 =1

Center at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−2 ),vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,−2 ),focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,−2 )

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

Approximately 35.71 feet

The Hyperbola

For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> y+1 ) 2 16 − ( x−4 ) 2 36 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> y+1 ) 2 4 2 − ( x−4 ) 2 6 2 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,−1 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,3 ),( 4,−5 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,−1+2 13 ),( 4,−1−2 13 )

<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 −4 x 2 +54y−16x+29=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 − y 2 −12x−6y−9=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−2 ) 2 2 2 − ( y+3 ) 2 ( 2 3 ) 2 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−3 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,−3 ),( 0,−3 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6,−3 ),( −2,−3 )

For the following exercises, graph the hyperbola, labeling vertices and foci.

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

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

 

 

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 −4 y 2 +6x+32y−91=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> y 2 − x 2 −12y−6=0

 

 

 

For the following exercises, find the equation of the hyperbola.

Center at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ),vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,4 ),focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,−6 )

Foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,7 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,7 ),vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6,7 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−5 ) 2 1 − ( y−7 ) 2 3 =1

The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

<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 =12x

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> x+2 ) 2 = 1 2 ( y−1 ); vertex:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,1 ); focus:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2, 9 8 ); directrix:<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> 7 8

<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 −6y−6x−3=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−y+23=0

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

For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

<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 +4y=0

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

 

 

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 −8x−10y+46=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> y 2 +12y+6x+15=0

 

 

 

For the following exercises, write the equation of the parabola using the given information.

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

Focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2, 9 8 ); directrix is<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> 7 8

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

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

<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 +24x−60y−60=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><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +14xy+5 y 2 +18x−6y+30=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 +xy+2 y 2 +8x−26y+9=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=−31<0, ellipse

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 +4xy−2 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 −xy+ y 2 −6=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 ∘ , x ′ 2 +3 y ′ 2 −12=0

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><mn>9</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −24xy+16 y 2 −80x−60y+100=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 −xy+ y 2 −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> 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 +24xy− y 2 −12x+26y+11=0

Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 1−5 cos θ

Hyperbola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>5</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and directrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>units to the left of the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 3+2 cos θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 4+3 sin θ

Ellipse with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 4  and directrix<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> 1 3  unit above the pole.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5−5 sin θ

For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 1−sin θ

 

 

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 4+3 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 4+5 cos θ

 

 

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 9 3−6 cos θ

For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is<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><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 1+cos  θ

Directrix is<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><mn>−2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and eccentricity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics></math>

Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 3 2 + y 2 2 2 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); foci:<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> 5 ,0 ),( − 5 ,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> y 2 +16 x 2 −36y+32x−92=0

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−3 ) 2 64 + ( y−2 ) 2 36 =1

center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,2 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3+2 7 ,2 ),( 3−2 7 ,2 )

 

 

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

Write the standard form equation of an ellipse with a center at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,2 ),vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,2 ),and focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> x−1 ) 2 36 + ( y−2 ) 2 27 =1

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

<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 49 − y 2 81 =1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 7 2 − y 2 9 2 =1; center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ); vertices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,0 ),( −7,0 ); foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 130 ,0 ),( − 130 ,0 ); asymptotes:<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><mo>±</mo><mfrac/></mrow></annotation-xml></semantics></math> 9 7 x

<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 −9 x 2 +128y+112=0

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

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

center:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−3 ); vertices:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 8,−3 ),( −2,−3 );foci:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3+ 26 ,−3 ),( 3− 26 ,−3 ); asymptotes:<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><mo>±</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 5 (x−3)−3

<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 +4y−4x−18=0

Write the standard form equation of a hyperbola with foci at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,0 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,6 ),and a vertex at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1,2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> y−3 ) 2 1 − ( x−1 ) 2 8 =1

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

<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 +10x=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 −12x−y+11=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> x−2 ) 2 = 1 3 ( y+1 ); vertex:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−1 ); focus:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,− 11 12 ); directrix:<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><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 13 12

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

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

<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 +8x−8y+40=0

 

 

 

Write the equation of a parabola with a focus at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,3 ) and directrix<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><mn>−1.</mn></mrow></annotation-xml></semantics></math>

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximately<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>8.49</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>feet

For the following exercises, determine which conic section is represented by the given equation, and then 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.

<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 −2xy+3 y 2 =4

<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 +6x−8y=0

parabola;<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><msup/></mrow></annotation-xml></semantics></math> 63.4 ∘

For the following exercises, rewrite 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, and graph the rotated graph.

<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 =4

<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 −125x=0

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

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2−sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 4+6 cos θ

Hyperbola with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 , and directrix<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> 5 6  units to the right of the pole.

For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 4−8 sin θ

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 4+4 sin θ

Find a polar equation of the conic with focus at the origin, eccentricity of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mn>2</mn><mo>,</mo></mrow></annotation-xml></semantics></math>and directrix:<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><mn>3.</mn></mrow></annotation-xml></semantics></math>

Glossary

eccentricity
the ratio of the distances from a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on the graph to the focus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>F</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and to the directrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represented by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> PF PD ,where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>e</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a positive real number
polar equation
an equation of a curve in polar coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</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>θ</mi></mrow></annotation-xml></semantics></math>