Let $d_1$ and $d_2$ be the lengths of perpendiculars drawn from foci $S$and $S$' of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to the tangent at any point P on the ellipse. Then,$SP : SP\text{'}=$

Let $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ be a tangent to the ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$  at point $P(a\cos \theta ,b\sin \theta )$ Then

$d_1=\mid \frac{e\cos \theta -1}{\sqrt{\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}}}\text{ and }{d}_2=\left|\frac{e\cos \theta +1}{\sqrt{\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}}}\right|$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{d}_1:d_2=|e\cos \theta -1|:|e\cos \theta +1|\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{d}_1:d_2=|ae\cos \theta -a|:|ae\cos \theta +a|\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{d}_1:d_2=SP:S^{\mathrm{\prime }}P.\end{array}$