Equation of angle bisector for
$a x^{2}+2 h x y +b y^{2}=0$
$\Rightarrow \frac{x^{2}-y^{2}}{a-b}=\frac{x y}{h}$
and for $x^{2}-2 p x y-y^{2}=0$
$\Rightarrow \frac{x^{2}-y^{2}}{1-(-1)}=\frac{x y}{-p}$
$\Rightarrow x^{2}-y^{2}+\frac{2 x y}{p}=0$
Given equation of angle bisector is
$x^{2}-2 q x y +y^{2}=0$
On comparing $\Rightarrow \frac{2}{P}=-2 q$
$\Rightarrow p q=-1$