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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Sin(A+B) = sinBcosA+cosBsinA
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192
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Practice Key Terms 2

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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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