Find a polar equation for the hyperbola with its focus at the pole, eccentricity $e = \frac { 7 } { 6 }$ and directrix $y = 7$.
A) $r = \frac { 49 } { 6 + 7 \sin \theta }$
B) $r = \frac { - 49 } { 6 + 7 \sin \theta }$
C) $r = \frac { 49 } { 6 - 7 \cos \theta }$
D) $r = \frac { - 49 } { 6 - 7 \cos \theta }$
E) $r = \frac { - 49 } { 6 - 7 \sin \theta }$

Question

Quizplus · Accepted Answer

The polar equation for a conic section with a focus at the pole is given by $r = \frac { ed } { 1 + e \sin \theta }$ or $r = \frac { ed } { 1 - e \cos \theta }$, where $e$ is the eccentricity and $d$ is the perpendicular distance from the directrix to the pole. Since the directrix is $y = 7$, $d = 7$, and the eccentricity $e = \frac { 7 } { 6 }$, the correct equation is $r = \frac { 49 } { 6 + 7 \sin \theta }$ because it matches the form $r = \frac { ed } { 1 + e \sin \theta }$ with the given values.

Find a Polar Equation for the Hyperbola with Its Focus e=76e = \frac { 7 } { 6 }e=67​

Find a Polar Equation for the Hyperbola with Its Focus $e = \frac { 7 } { 6 }$