Find a polar equation of the conic with the given characteristics and with one focus at the pole.
$\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \ \text { Hyperbola } & e = 5 & x = 2 \end{array}$
A) $r = \frac { 10 } { 1 - 5 \cos \theta }$
B) $r = \frac { 10 } { 1 + 5 \cos \theta }$
C) $r = \frac { 2 } { 1 + 5 \cos \theta }$
D) $r = \frac { 10 } { 1 + 5 \sin \theta }$
E) $r = \frac { 10 } { 1 - 5 \sin \theta }$

Question

Quizplus · Accepted Answer

The general polar equation for a conic section with one focus at the pole is $r = \frac{ed}{1 \pm e\cos\theta}$ or $r = \frac{ed}{1 \pm e\sin\theta}$, where $e$ is the eccentricity and $d$ is the perpendicular distance from the directrix to the pole. For a hyperbola with eccentricity $e = 5$ and directrix $x = 2$, the distance $d = 2$. Since the directrix is vertical (x = constant), we use the cosine form. The equation becomes $r = \frac{10}{1 + 5\cos\theta}$, as the hyperbola opens away from the directrix.

Find a Polar Equation of the Conic with the Given Conic Eccentricity Directrix Hyperbola e=5x=2\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e = 5 & x = 2 \end{array} Conic Hyperbola ​ Eccentricity e=5​ Directrix x=2​

Find a Polar Equation of the Conic with the Given $\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e = 5 & x = 2 \end{array}$