Let m be the slope of the lines $ax^2 +2 hxy + by^2 = 0$, then according to question , other slope will be nm.
$\therefore m+ nm=\frac{-2h}{b}$
$\Rightarrow m \left(1+n\right)=\frac{-2h}{b}\quad\quad... \left(i\right)$
and$\quad m ยท nm= \frac{a}{b}$
$\Rightarrow nm^{2} = \frac{a}{b}$
$\Rightarrow m = \pm\sqrt{\frac{a}{bn}} \quad\quad... \left(ii\right)$
$\therefore $ From Eq. (i), we get
$\pm\sqrt{\frac{a}{bn}}\left(1+n\right) = \frac{-2h}{b}$
On squaring both sides, we get
$\frac{a}{bn}\left(1+n\right)^{2} = \frac{4h^{2}}{b^{2}}$
$\Rightarrow 4h^{2}n = ab\left(1+n\right)^{2}$