If $CF$ is perpendicular from the centre of the ellipse

$\frac{x^2}{25}+\frac{y^2}{9}=1$ to the tangent at $P, G$ is the point where

the normal at P meets the major axis, then $CF. PG=$

Let $P(5\cos \theta ,3\sin \theta )$

Equation of the tangent at $P$ is

$\begin{array}{l}\frac{x\cos \theta }{5}+\frac{y\sin \theta }{3}=1\\ \Rightarrow CF=\frac{1}{\sqrt{\frac{{\cos }^2 \theta }{25}+\frac{{\sin }^2 \theta }{9}}}=\frac{15}{\sqrt{9{\cos }^2 \theta +25{\sin }^2 \theta }}\end{array}$

Equation of the normal at $P$ is

$\frac{5x}{\cos \theta }-\frac{3y}{\sin \theta }=25-9=16$

Coordinates of $G$ are $\left(\frac{16\cos \theta }{5},0\right)$

$\Rightarrow PG=\frac{3}{5}\sqrt{9{\cos }^2 \theta +25{\sin }^2 \theta }$