Let P be a point on the ellipse x ^ 2 9 + y ^ 2 4 = 1 . Let the...

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

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

Let, $R ≡ \left(h, k\right)$

$⇒ h = 3cos\theta, k = \dfrac{18}{7}sin\theta$

$⇒ \dfrac{h}{3} = cos\theta, \dfrac{7k}{18} = sin\theta$

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

$⇒ e = \sqrt{1 - \dfrac{324}{49 \times 9}}$

$⇒ e = \dfrac{\sqrt{117}}{21}$

$⇒ e = \dfrac{\sqrt{13}}{7}$