Let P ( 2 3 7 , ? 6 7 ) , Q, R and S be four points on the ellipse 9 x 2 + 4 y 2 = 36 . Let PQ and RS be mutually perpendicular and pass through the origin. If 1 ( P Q ) 2 + 1 ( R S ) 2 = p q , where p and q are coprime, then p + q is equal to

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Solution

Q.
Let $P (\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}})$ , Q, R and S be four points on the ellipse $9 x^{2} + 4 y^{2} = 36$ . Let PQ and RS be mutually perpendicular and pass through the origin. If $\frac{1}{{(P Q)}^{2}} + \frac{1}{{(R S)}^{2}} = \frac{p}{q},$ where $p$ and $q$ are coprime, then $p + q$ is equal to [2023]

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