Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 3 } { 2 }$ , and directrix $x = - 6$ . Identify the conic.
A) $r = \frac { 18 } { 2 - 3 \sin \theta }$ , hyperbola
B) $r = \frac { 18 } { 2 - 3 \cos \theta }$ , hyperbola
C) $r = \frac { 18 } { 2 - \cos \theta }$ , ellipse
D) $r = \frac { 18 } { 2 - \sin \theta }$ , ellipse

Question

Quizplus · Accepted Answer

The polar equation of a conic with focus at the origin is $ r = \frac{ed}{1 - e \cos \theta} $ for a directrix $ x = -d $. Given $ e = \frac{3}{2} $ and $ d = 6 $, the equation becomes $ r = \frac{18}{2 - 3 \cos \theta} $, which is a hyperbola since $ e > 1 $.

Write a Polar Equation of the Conic That Has a Focus