Find a polar equation for the ellipse with its focus at the pole, eccentricity $e = \frac { 5 } { 6 } , \text { and }$ directrix $y = - 10$.
A) $r = \frac { - 50 } { 6 + 5 \sin \theta }$
B) $r = \frac { - 50 } { 6 - 5 \cos \theta }$
C) $r = \frac { 50 } { 6 - 5 \sin \theta }$
D) $r = \frac { - 50 } { 6 + 5 \cos \theta }$
E) $r = \frac { 50 } { 6 - 5 \cos \theta }$

Question

Quizplus · Accepted Answer

The polar equation for an ellipse 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 $y = -10$, $d = 10$, and $e = \frac{5}{6}$, the correct equation must have $ed = \frac{5}{6} \times 10 = \frac{50}{6}$. The directrix being horizontal (y = constant) suggests the use of $\sin \theta$ in the denominator. Thus, the correct equation is $r = \frac { 50 } { 6 - 5 \sin \theta }$.

Find a Polar Equation for the Ellipse with Its Focus e=56, and e = \frac { 5 } { 6 } , \text { and }e=65​, and

Find a Polar Equation for the Ellipse with Its Focus $e = \frac { 5 } { 6 } , \text { and }$