The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section.
-$$e = \frac { 1 } { 5 } , x = - 10$$
A) $r = \frac { 10 } { 1 + 5 \cos \theta }$
B) $r = \frac { 10 } { 5 - \sin \theta }$
C) $r = \frac { 10 } { 5 + \cos \theta }$
D) $r = \frac { 10 } { 5 - \cos \theta }$

Question

Quizplus · Accepted Answer

Since we are given that the focus is at the origin (i.e., $F(0,0)$), the polar equation of the conic section is of the form $r = \frac{ed}{1-e\cos	heta}$, where $e$ is the eccentricity and $d$ is the distance from the origin to the directrix. We are given that $e = \frac{1}{5}$ and that the directrix is $x = -10$. Since the directrix is a vertical line, the distance from the origin to the directrix is simply $d = 10$. Thus, the polar equation of the conic section is:
$$r = \frac{\frac{1}{5}\cdot 10}{1 - \frac{1}{5}\cos	heta} = \boxed{\frac{10}{5 - \cos	heta}}.$$

The Eccentricity Is Given of a Conic Section with One e=15,x=−10e = \frac { 1 } { 5 } , x = - 10e=51​,x=−10

The Eccentricity Is Given of a Conic Section with One $e = \frac { 1 } { 5 } , x = - 10$