Let d₁ and d₂ be the length of the perpendiculars drawn from 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 =

$Given ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Focus are S=(ae,0) and S'=(-ae,0)

Tangent at $\mathrm{P}(\mathrm{acos} \mathrm{\alpha },\mathrm{bsin} \mathrm{\alpha })$ is
$\frac{\mathrm{x}}{\mathrm{a}}\cos \mathrm{\alpha }+\frac{\mathrm{y}}{\mathrm{b}}\sin \mathrm{\alpha }=1$          ….(1)
Distance of focus S(ae, 0) from this tangent is
${\mathrm{d}}_1=\frac{|\mathrm{ecos}\mathrm{\alpha }-1|}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}}$
Distance of focus S’(-ae ,0) from this line

$$\begin{gathered} {\mathrm{d}}_2=\frac{1+\mathrm{ecos}\mathrm{\alpha }}{\sqrt{\frac{{\cos }^2\mathrm{\alpha }}{{\mathrm{a}}^2}+\frac{{\sin }^2\mathrm{\alpha }}{{\mathrm{b}}^2}}} \\ Now \frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{1+\mathrm{ecos}\mathrm{\alpha }} \end{gathered}$$ 
We know that $\mathrm{SP}=\mathrm{a}-\mathrm{aecos}\mathrm{\alpha }\text{ and }{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}=\mathrm{a}+\mathrm{aecos} \mathrm{\alpha }$ 
$\Rightarrow \frac{\mathrm{SP}}{{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}}=\frac{1-\mathrm{ecos}\mathrm{\alpha }}{1+\mathrm{ecos}\mathrm{\alpha }}\Rightarrow \frac{\mathrm{SP}}{{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}}=\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}.$