Let d₁, and d₂ be the lengths of the perpendiculars drawn from the foci S and S' of the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ to the tangent at any point P on the ellipse.   Then, SP : S'P =

Tangent at $\mathrm{P}(\mathrm{acos}\mathrm{\alpha },\mathrm{bsin}\mathrm{\alpha })$ is $\frac{\mathrm{x}}{\mathrm{a}}\cos \mathrm{\alpha }+\frac{\mathrm{y}}{\mathrm{b}}\sin \mathrm{\alpha }=1$
The distance of focus s(ae,0) from this tangent is
${\mathrm{d}}_1=\frac{|\mathrm{ecos}\mathrm{\alpha }-1|}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}}$
' *u distance of focus s'(-ae, 0) from this line is ${\mathrm{d}}_2=\frac{1+\mathrm{ecos}\mathrm{\alpha }}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}}$

or  $\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{1+\mathrm{ecos}\mathrm{\alpha }}$

Now, $\mathrm{SP}=\mathrm{a}-\mathrm{aecos}\mathrm{\alpha }\text{ and }{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}=\mathrm{a}+\mathrm{aecos}\mathrm{\alpha }$

or  $\frac{\mathrm{SP}}{S^{\text{'}}P}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{1+\mathrm{ecos}\mathrm{\alpha }}=\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}$