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.
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.