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

equation  of chord of contact of $\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1)$ is

$\frac{\mathrm{x} {\mathrm{x}}_1}{{\mathrm{a}}^2}+\frac{\mathrm{y} {\mathrm{y}}_1}{{\mathrm{b}}^2}=1$

slope of line ${\mathrm{m}}_1=\frac{-{\mathrm{b}}^2{\mathrm{x}}_1}{{\mathrm{a}}^2{\mathrm{y}}_1}$

equation  of chord of contact of $\mathrm{P}({\mathrm{x}}_2,{\mathrm{y}}_2)$ is

$\frac{\mathrm{x} {\mathrm{x}}_2}{{\mathrm{a}}^2}+\frac{\mathrm{y} {\mathrm{y}}_2}{{\mathrm{b}}^2}=1$

slope of line ${\mathrm{m}}_2=\frac{-{\mathrm{b}}^2{\mathrm{x}}_2}{{\mathrm{a}}^2{\mathrm{y}}_2}$

$$\begin{gathered} {\mathrm{m}}_1{\mathrm{m}}_2=-1 \\ \frac{{\mathrm{b}}^4 {\mathrm{x}}_1{\mathrm{x}}_2}{{\mathrm{a}}^4\times {\mathrm{y}}_1{\mathrm{y}}_2}=-1 \\ \frac{{\mathrm{x}}_1 {\mathrm{x}}_2}{{\mathrm{y}}_1 {\mathrm{y}}_2}=-\frac{{\mathrm{a}}^4}{{\mathrm{b}}^4} \end{gathered}$$