Question

Find the condition for the line $lx+my+n=0$ is normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .

Answer 1

As we know that

Equation of normal with slope $p$ to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $y=px\pm \frac{p(a^2+b^2)}{\sqrt{a^2-b^2{p}^2}}$

Given $lx+my+n=0$ is the normal

$⟹y=-\frac{l}{m}x-\frac{n}{m}$

$⟹p=-\frac{l}{m},$

And also

$\pm \frac{p(a^2+b^2)}{\sqrt{a^2-p^2{b}^2}}=-\frac{n}{m}$

Squaring on both sides

$⟹p^2(a^2+b^2)=\frac{n^2}{m^2}(a^2-p^2{b}^2)$

$⟹\frac{l^2}{m^2}(a^2+b^2)=\frac{n^2}{m^2}(a^2-\frac{l^2}{m^2}{b}^2)$

$⟹l^2{m}^2(a^2+b^2)=n^2(a^2{m}^2-l^2{b}^2)$