Let P be a point on the ellipse x 2 9 + y 2 4 = 1 . Let the line passing through P and parallel to y - axis…

Given equation of ellipse and circle are $\frac{x^2}{9}+\frac{{\displaystyle y^2}}{4}=1$ and $x^2+y^2=9$.

$\Rightarrow P\equiv \left(3\cos \theta , 2\sin \theta \right) \text{and} Q\equiv \left(3\cos \theta , 3\sin \theta \right)$

Let, $R\equiv \left(h,k\right)$

$\Rightarrow h=3\cos \theta , k=\frac{18}{7}\sin \theta$

$\Rightarrow \frac{h}{3}=\cos \theta , \frac{7k}{18}=\sin \theta$

So, locus of $R$ is given by, $\frac{x^2}{9}+\frac{49y^2}{324}=1$.

$\Rightarrow e=\sqrt{1-\frac{324}{49\times 9}}$

$\Rightarrow e=\frac{\sqrt{117}}{21}$

$\Rightarrow e=\frac{\sqrt{13}}{7}$