$\mathrm{P}\left(\mathrm{\theta }\right)$ and $\mathrm{Q}=\left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right)$ are two variable points on $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1.$ Then locus of mid point of PQ is $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\mathrm{K}$ the K=

Given ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

Given $\mathrm{P}\left(\mathrm{\theta }\right), \mathrm{Q}\left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right)$ are two points on ellipse then 

$\begin{array}{rcl}\mathrm{P}& =& (\mathrm{a} \mathrm{cos\theta }, \mathrm{b} \mathrm{sin\theta }), \mathrm{Q}=\left(\mathrm{a} \cos \left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right), \mathrm{b} \sin \left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right)\right)\\ & \Rightarrow & \mathrm{P}=(\mathrm{a} \mathrm{cos\theta }, \mathrm{b} \mathrm{sin\theta }), \mathrm{Q}=(-\mathrm{a} \mathrm{sin\theta }, \mathrm{b} \mathrm{cos\theta }))\end{array}$

Let $\mathrm{C} (\mathrm{x}, \mathrm{y})$ be locus of mid point of $\overline{\mathrm{PQ}}$

$\begin{array}{rcl} & \Rightarrow & (\mathrm{x}, \mathrm{y})=\left(\frac{\mathrm{a} \mathrm{cos\theta }-\mathrm{a} \mathrm{sin\theta }}{2}, \frac{\mathrm{b} \mathrm{sin\theta }+\mathrm{b} \mathrm{cos\theta }}{2}\right)\\ & \Rightarrow & \mathrm{x}=\mathrm{a}\left[\frac{\mathrm{cos\theta }-\mathrm{sin\theta }}{2}\right], \mathrm{y}=\mathrm{b}\left(\frac{\mathrm{sin\theta }+\mathrm{cos\theta }}{2}\right)\\ & \Rightarrow & \frac{\mathrm{cos\theta }-\mathrm{sin\theta }}{2}=\frac{\mathrm{x}}{\mathrm{a}} -\left(1\right), \frac{\mathrm{sin\theta }+\mathrm{cos\theta }}{2}=\frac{\mathrm{y}}{\mathrm{b}} -\left(2\right)\\ {\left(1\right)}^2+{\left(2\right)}^2 & \Rightarrow & {\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2+{\left(\frac{\mathrm{y}}{\mathrm{b}}\right)}^2={\left(\frac{\mathrm{cos\theta }-\mathrm{sin\theta }}{2}\right)}^2+{\left(\frac{\mathrm{cos\theta }+\mathrm{sin\theta }}{2}\right)}^2\\ & \Rightarrow & \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\frac{2({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta })}{4}\\ & \Rightarrow & \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\frac{1}{2}\end{array}$

Given locus of mid point of $\overline{\mathrm{PQ}}$ is 

$\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\mathrm{k}$

By Comparision $\mathrm{k}=\frac{1}{2}$