Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic VerticesEllipse

Question

Quizplus · Accepted Answer

For an ellipse with one focus at the pole, the polar equation is given by:
$r(	heta) = \dfrac{d}{1-e\cos	heta}$
where $d$ is the distance between the focus and the origin (in this case, the vertex of the ellipse) and $e$ is the eccentricity. 
Since the vertices of the ellipse are at a distance of 3 from the origin (the pole), we have $d=3$. 
Also, the distance between the center and one of the vertices of an ellipse is given by $a$, where $a$ is the major axis. Since the major axis is twice the distance between the vertices, we have $a=2	imes 11=22$. 
Similarly, the distance between the center and one of the foci of an ellipse is given by $c$, where $c=\sqrt{a^2-b^2}$, where $b$ is the minor axis. Since the foci are at a distance of 3 from the origin, we have $c=3$. Therefore, we can solve for $b$:
$b = \sqrt{a^2-c^2} = \sqrt{22^2-3^2} = \sqrt{463}$.
Finally, we can find the eccentricity:
$e = \dfrac{c}{a} = \dfrac{3}{22}$.
Substituting these values in the polar equation, we get:
$r(	heta) = \dfrac{3}{1-\frac{3}{22}\cos	heta}$.
Simplifying, we get:
$r(	heta) = \dfrac{66}{22-3\cos	heta}$.
Therefore, the answer is choice C.

Find a Polar Equation of the Conic with the Given