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 polar equation of a conic section with eccentricity $e$ and directrix $x = d$ (assuming the focus is at the origin) is given by $r = \frac{ed}{1 \pm e\cos\theta}$ for horizontal directrices and $r = \frac{ed}{1 \pm e\sin\theta}$ for vertical directrices. Given a hyperbola with eccentricity $e = 5$ and directrix $x = 2$, the equation becomes $r = \frac{10}{1 + 5\cos\theta}$, since the directrix is vertical and to the right of the focus, indicating a positive cosine in the denominator.

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 $\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e = 5 & x = 2 \end{array}$