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
$r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{and}y=r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta .$
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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
$\text{\hspace{0.17em}}P(r,\theta )\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{}\mathrm{cos}\text{}\theta ,$ and
$\text{\hspace{0.17em}}y=r\text{}\mathrm{sin}\text{}\theta \text{\hspace{0.17em}}$ 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, and the denominator involves
$\text{\hspace{0.17em}}\mathrm{sin}\text{}\theta ,$ what conclusion can be drawn about the directrix?
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.