Find the polar equation of the conic with focus at the pole, having the given properties. Identify the conic.
-Directrix $x = - 8$, eccentricity 3
A) $r = \frac { 24 } { 1 - 3 \cos \theta }$, hyperbola
B) $r = \frac { 24 } { 1 - 8 \cos \theta }$, hyperbola
C) $r = \frac { 8 } { 8 + \cos \theta }$, ellipse
D) $r = \frac { 3 } { 8 - 8 \cos \theta }$, parabola

Question

Quizplus · Accepted Answer

The polar equation of a conic section with a focus at the pole is given by $r = \frac { ed } { 1 - e \cos \theta }$, where $e$ is the eccentricity and $d$ is the perpendicular distance from the directrix to the pole. Given $x = -8$ as the directrix and an eccentricity of 3, $d = 8$ and $e = 3$, making the equation $r = \frac { 24 } { 1 - 3 \cos \theta }$. Since the eccentricity is greater than 1, the conic is a hyperbola.

Find the Polar Equation of the Conic with Focus at the Pole