Question

$C:x^2+y^2=9$, $E:\frac{x^2}{9}+\frac{y^2}{4}=1$, $L:y=2x$

$P$ is a point on the circle $C$. The perpendicular $PQ$ to the major axis of the ellipse $E$ meets the ellipse at $M$. Then $\frac{MQ}{PQ}$ is equal to

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{2}$
• D. none of these

Answer 1

The correct option is B $\frac{2}{3}$

Let the coordinates of $P$ be $(3\cos \theta ,3\sin \theta )$. Then the eccentric angle of $M$, the point where the ordinate $PQ$ through $P$ meets the ellipse is $\theta$ and the coordinates of $M$ are $(3\cos \theta ,2\sin \theta )$, $PQ=3\sin \theta$.
$MQ=2\sin \theta$ so $\frac{MQ}{PQ}=\frac{2}{3}$