# 12.5 Conic sections in polar coordinates  (Page 4/8)

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## Converting a conic in polar form to rectangular form

Convert the conic $\text{\hspace{0.17em}}r=\frac{1}{5-5\mathrm{sin}\text{\hspace{0.17em}}\theta }$ to rectangular form.

We will rearrange the formula to use the identities

Convert the conic to rectangular form.

$4-8x+3{x}^{2}-{y}^{2}=0$

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 $\text{\hspace{0.17em}}P\left(r,\theta \right)\text{\hspace{0.17em}}$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
• A conic is the set of all points $\text{\hspace{0.17em}}e=\frac{PF}{PD},$ where eccentricity $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ 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 and to convert the equation for a conic from polar to rectangular form. See [link] .

## 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 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.

Parabola with $\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{3}{4}\text{\hspace{0.17em}}$ units below the pole.

Hyperbola with $\text{\hspace{0.17em}}e=2\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{5}{2}\text{\hspace{0.17em}}$ units above the pole.

Parabola with $\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{3}{10}\text{\hspace{0.17em}}$ units to the right of the pole.

Ellipse with $\text{\hspace{0.17em}}e=\frac{2}{7}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ units to the right of the pole.

Hyperbola with $\text{\hspace{0.17em}}e=\frac{5}{3}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{11}{5}\text{\hspace{0.17em}}$ units above the pole.

Hyperbola with $\text{\hspace{0.17em}}e=\frac{8}{7}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{7}{8}\text{\hspace{0.17em}}$ units to the right of the pole.

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