Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$.
A)
$r = \frac { 84 } { 1 - 84 \sin \theta }$
B) $r = \frac { 84 } { 1 - 7 \sin \theta }$
C) $r = \frac { 84 } { 1 + 7 \sin \theta }$
D) $r = \frac { 84 } { 1 - \sin \theta }$
E) $r = \frac { 12 } { 1 + 12 \sin \theta }$

Question

Quizplus · Accepted Answer

Since the focus is at the origin, we have the equation of the hyperbola in polar coordinates as 
$$r = \frac{	ext{eccentricity}}{1-	ext{eccentricity}\cos	heta}$$
where $	ext{eccentricity}=7$. 
The distance from any point $(r,	heta)$ on the hyperbola to the directrix $r=-12\csc	heta$ is given by 
$$\frac{\left|r+\frac{12}{\sin	heta}ight|}{\sqrt{1+\left(\frac{12}{r\sin	heta}ight)^2}}=\frac{r^2-12r\cos	heta}{7}$$
Simplifying this equation gives
$$r= \frac{84}{7\sin	heta-r\cos	heta}= \frac{84}{7\sin	heta-\sqrt{r^2-84^2\sin^2	heta}}$$
Squaring both sides and simplifying gives us
$$r^2=\frac{7056}{1-49\sin^2	heta}$$
which is equivalent to
$$r = \frac{84}{\sqrt{1-7^2\sin^2	heta}}$$
Thus, the correct polar equation is $r = \frac { 84 } { 1 - 7 \sin 	heta }$.

Write a Polar Equation In r\mathrm { r }r And θ\thetaθ Of a Hyperbola with the Focus at the Origin, with \theta

Write a Polar Equation In $\mathrm { r }$ And $\theta$ Of a Hyperbola with the Focus at the Origin, with \theta