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Converting a conic in polar form to rectangular form

Convert the conic r = 1 5 5 sin θ to rectangular form.

We will rearrange the formula to use the identities   r = x 2 + y 2 , x = r cos θ , and  y = r sin θ .

                           r = 1 5 5 sin θ   r ( 5 5 sin θ ) = 1 5 5 sin θ ( 5 5 sin θ ) Eliminate the fraction .         5 r 5 r sin θ = 1 Distribute .                          5 r = 1 + 5 r sin θ Isolate  5 r .                      25 r 2 = ( 1 + 5 r sin θ ) 2 Square both sides .           25 ( x 2 + y 2 ) = ( 1 + 5 y ) 2 Substitute  r = x 2 + y 2  and  y = r sin θ .         25 x 2 + 25 y 2 = 1 + 10 y + 25 y 2 Distribute and use FOIL .           25 x 2 10 y = 1 Rearrange terms and set equal to 1 .
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Convert the conic r = 2 1 + 2   cos   θ to rectangular form.

4 8 x + 3 x 2 y 2 = 0

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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 P ( r , θ ) at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • A conic is the set of all points e = P F P D , where eccentricity e 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 r = x 2 + y 2 , x = r   cos   θ , and y = r   sin   θ 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.

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

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If a conic section is written as a polar equation, and the denominator involves sin   θ , what conclusion can be drawn about the directrix?

The directrix will be parallel to the polar axis.

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

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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.

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Algebraic

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

r = 6 1 2   cos   θ

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r = 3 4 4   sin   θ

Parabola with e = 1 and directrix 3 4 units below the pole.

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r = 8 4 3   cos   θ

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r = 5 1 + 2   sin   θ

Hyperbola with e = 2 and directrix 5 2 units above the pole.

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r = 16 4 + 3   cos   θ

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r = 3 10 + 10   cos   θ

Parabola with e = 1 and directrix 3 10 units to the right of the pole.

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r = 4 7 + 2   cos   θ

Ellipse with e = 2 7 and directrix 2 units to the right of the pole.

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r ( 1 cos   θ ) = 3

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r ( 3 + 5 sin   θ ) = 11

Hyperbola with e = 5 3 and directrix 11 5 units above the pole.

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r ( 4 5 sin   θ ) = 1

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r ( 7 + 8 cos   θ ) = 7

Hyperbola with e = 8 7 and directrix 7 8 units to the right of the pole.

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Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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