$\text{ The product of the perpendicular from any point on the hyperbola }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1\text{ to its asymptotes is }$

$$\begin{gathered} Given hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ Let P=(asec\theta ,b \tan \theta ) be any point on hyperbola \\ Equations of asymptotes are bx+ay=0\to \left(1\right) and bx-ay=0\to \left(2\right) \\ Let d_1,d_2 be perpendicular dis\tan ces from P to \left(1\right) and \left(2\right) respectively \\ Now d_1{d}_2=\frac{ab sec\theta +ab \tan \theta }{\sqrt{a^2+b^2}}\left(\frac{ab sec\theta -ab \tan \theta }{\sqrt{a^2+b^2}}\right) \\ =\frac{a^2{b}^2}{a^2+b^2} \end{gathered}$$

Product of perpendiculars from any point on hyperbola to its asymptotes is $\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2+{\mathrm{b}}^2}$