Let P be a point on the ellipse x^29+y^24=1.

Question

Let $P$ be a point on the ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle ${x}^{2}+{y}^{2}=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse, is :

Accepted Answer

$\begin{aligned} & \mathrm{h}=3 \cos 	heta \\ & \mathrm{k}=\frac{18}{7} \sin 	heta\end{aligned}$

$\begin{aligned} & 	herefore 	ext { locus }=\frac{\mathrm{x}^2}{9}+\frac{49 \mathrm{y}^2}{324}=1 \\ & \mathrm{e}=\sqrt{1-\frac{324}{49 	imes 9}}=\frac{\sqrt{117}}{21}=\frac{\sqrt{13}}{7}\end{aligned}$

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