Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$.
A) $r = \frac { 84 } { 1 - 84 \sin \theta }$
B) $r = \frac { 84 } { 1 - 7 \sin \theta }$
C) $r = \frac { 84 } { 1 + 7 \sin \theta }$
D) $r = \frac { 84 } { 1 - \sin \theta }$
E) $r = \frac { 12 } { 1 + 12 \sin \theta }$

Question

Quizplus · Accepted Answer

The polar equation of a conic section with a focus at the origin is given by $r = \frac { ed } { 1 - e \sin \theta }$ for a hyperbola, where $e$ is the eccentricity and $d$ is the perpendicular distance from the directrix to the origin. Given the eccentricity $e = 7$ and the directrix $r = -12 \csc \theta$, which implies $d = 12$, the correct equation is $r = \frac { 84 } { 1 - 7 \sin \theta }$.

Write a Polar Equation In r\mathrm { r }r And θ\thetaθ Of a Hyperbola with the Focus at the Origin, with \theta

Write a Polar Equation In $\mathrm { r }$ And $\theta$ Of a Hyperbola with the Focus at the Origin, with \theta