If the normal at any point $P$ on the ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis at $G_1$ and the minor axis at $G_2$, then

Let the coordinates of $P$ be $(a \cos \theta , b \sin \theta )$ Equation of the normal at $P$ is

$ax \sec \theta -by \mathrm{cosec} \theta =a^2-b^2=a^2{e}^2$

It meets the major axis at $G_1\left(a{e}^2\cos \theta , 0\right)$ and minor axis at $G_2\left(0, -\frac{a^2{e}^2}{b}\sin \theta \right)$ 

So

$\begin{array}{l}{\left(P{G}_1\right)}^2={\left(a\cos \theta -a{e}^2\cos \theta \right)}^2+b^2{\sin }^2\theta \\ =a^2{\cos }^2\theta {\left(1-e^2\right)}^2+b^2{\sin }^2\theta \\ =a^2{\cos }^2\theta {\left(\frac{b^2}{a^2}\right)}^2+b^2{\sin }^2\theta \\ =\frac{b^2}{a^2}\left(b^2{\cos }^2\theta +a^2{\sin }^2\theta \right)\end{array}$

and 

$$\begin{gathered} \begin{array}{l}{\left(P{G}_2\right)}^2=(a\cos \theta {)}^2+{\left(b+\frac{a^2{e}^2}{b}\right)}^2{\sin }^2\theta \\ =(a\cos \theta {)}^2+\frac{a^4}{b^2}{\sin }^2 \theta \\ =\frac{a^2}{b^2}\left(b^2{\cos }^2 \theta +a^2{\sin }^2 \theta \right)\end{array} \\ P{G}_1 : P{G}_2=b^2 : a^2 \end{gathered}$$