Find a polar equation for the ellipse with its focus at the pole and vertices $( 6,0 ) , ( 24 , \pi )$
A) $r = \frac { 48 } { 5 + 3 \cos \theta }$
B) $r = \frac { 24 } { 3 + 2 \cos \theta }$
C) $r = \frac { 48 } { 3 + 2 \cos \theta }$
D) $r = \frac { 48 } { 3 + 2 \sin \theta }$
E) $r = \frac { 24 } { 5 + 3 \cos \theta }$

Question

Quizplus · Accepted Answer

The correct polar equation for an ellipse with a focus at the pole and given vertices is derived using the formula $r = \frac{l}{1 - e\cos\theta}$, where $l$ is the semi-latus rectum and $e$ is the eccentricity. Given vertices at $6,0$ and $24,\pi$, the major axis length is $30$ (distance between vertices), and the semi-major axis length $a = 15$. The distance from the center to a focus (c) is $9$ (difference between semi-major axis and distance to closer vertex), so the eccentricity $e = c/a = 9/15 = 3/5$. The semi-latus rectum $l = b^2/a$, where $b$ is the semi-minor axis. Using the ellipse equation $b^2 = a^2(1-e^2)$, we find $b^2 = 225(1-(3/5)^2) = 225(1-9/25) = 225(16/25) = 144$, so $l = 144/15 = 48/5$. Thus, the correct equation is $r = \frac{48}{5 + 3\cos\theta}$.

Find a Polar Equation for the Ellipse with Its Focus (6,0),(24,π)( 6,0 ) , ( 24 , \pi )(6,0),(24,π)

Find a Polar Equation for the Ellipse with Its Focus $( 6,0 ) , ( 24 , \pi )$