Parallelogram $PQSR$ is a cyclic quadrilateral
$\Rightarrow$ Parallelogram is a rectangle
$\Rightarrow \angle QPR={90}^{\mathrm{o}}$
Let, the equation of the tangent from $P\left(h,k\right)$ to the ellipse is $y=mx\pm \sqrt{16m^2+9}\Rightarrow {\left(k-mh\right)}^2=16m^2+9$
$\because m_1{m}_2=-1\Rightarrow h^2-16=-k^2+9$
$\Rightarrow$ locus of $P$ is $x^2+y^2=25$

Hence, the area of locus of $P$ is $25\pi$ sq. units