Question

Let d and d’ be the perpendicular distances from the foci of an ellipse to the tangent at P on the ellipse, whose foci are S and S’. Then S’P:SP =

Answer 1

$P\left(\theta \right)=(a\cos \theta ,b\sin \theta )$
$\frac{SP}{S^{\prime }P}=\frac{a(1-e\cos \theta )}{a(1+e\cos \theta )}$
From the similar triangles S’ QT and SRT,
We have $\frac{S^{\prime }Q}{SR}=\frac{S^{\prime }T}{ST}$
$\frac{d^{\prime }}{d}=\frac{\frac{a}{\cos \theta }+ae}{\frac{a}{\cos \theta }-ae}=\frac{a(1+e\cos \theta )}{a(1-e\cos \theta )}=\frac{S^{\prime }P}{SP}$
$\therefore$ $S^{\prime }P:SP=d^{\prime }:d$