If the intercepts made by tangent, normal to a rectangular ${\mathrm{x}}^2-{\mathrm{y}}^2={\mathrm{a}}^2$ with x–axis are ${\mathrm{a}}_1, {\mathrm{a}}_2$ and with y-axis are ${\mathrm{b}}_1,{\mathrm{b}}_2$ then ${\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2$=

$Given hyperbola x^2-y^2=a^2$

$Let P\left(\theta \right)=(asec\theta ,a\tan \theta ) be any point on given hyperbola$

 $$\begin{gathered} \mathrm{Equation} \mathrm{of} \mathrm{tangent} \mathrm{at} \mathrm{P}\left(\mathrm{\theta }\right) \mathrm{is} \mathrm{xsec}\mathrm{\theta }-\mathrm{ytan}\mathrm{\theta }=\mathrm{a} \\ \mathrm{It} \mathrm{meets} \mathrm{coordinate} \mathrm{axes} \mathrm{at} {\mathrm{a}}_{1 } \mathrm{and} {\mathrm{b}}_{1 }\mathrm{then} \end{gathered}$$

${\mathrm{a}}_1=\frac{\mathrm{a}}{\sec \mathrm{\theta }},{\mathrm{b}}_1=\frac{-\mathrm{a}}{\tan \mathrm{\theta }}$

Normal at $\mathrm{P}\left(\mathrm{\theta }\right) \mathrm{is} \frac{\mathrm{x}}{\sec \mathrm{\theta }}+\frac{\mathrm{y}}{\tan \mathrm{\theta }}=2\mathrm{a}$

$it meets coordinate axes at a_2,b_2 then$

$$\begin{gathered} {\mathrm{a}}_2=2\mathrm{asec}\mathrm{\theta },{\mathrm{b}}_2=2\mathrm{atan}\mathrm{\theta } \\ \mathrm{Now} {\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2=\frac{\mathrm{a}}{\mathrm{sec\theta }}(2\mathrm{a} \mathrm{sec\theta })+\frac{-\mathrm{a}}{\mathrm{tan\theta }}(2\mathrm{atan}\mathrm{\theta }) \\ =2{\mathrm{a}}^2 -2{\mathrm{a}}^2 =0 \end{gathered}$$