If $P( heta)$ and $Q\left( \frac{\pi}{2} + heta ight)$ are two points on the ellipse $\frac{x^{

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If $P(	heta)$ and $Q\left( \frac{\pi}{2} + 	heta ight)$ are two points on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, then locus of midpoint of $PQ$ is:

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Sol.(a) $2h = a(cos	heta - sin	heta)$$$2k = b(cos	heta + sin	heta)$$( $- asin	heta,bcos	heta$ )$$\begin{matrix} & \ \Rightarrow \left( \frac{2h}{a} ight)^{2} + \left( \frac{2k}{b} ight)^{2} = 2\left( \cos^{2}	heta + \sin^{2}	heta ight) \ & \ \Rightarrow \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{1}{2} \end{matrix}$$

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