If $m_{1}$ and $m_{2}$ are slopes of the lines represented by $ax^{2}+2hxy+by^{2}=0$, then
$m_{1}+m_{2}=-\frac{2h}{b}$ and $m_{1}m_{2}=\frac{a}{b}$
The given equation is
$x^{2}+2 hxy +2y^{2} =0$
On comparing this equation with
$ax^{2}+2hxy+by^{2}=0$, we get
$a = 1$, $2h = 2h$ and $b = 2$
Let the slopes of lines are $m_{1}$ and $m_{2}$
$\therefore m_{1}: m_{2}=1: 2$
Let $m_{1}=m$ and $m_{2}=2m$
$\therefore m_{1}+ m_{2}=-\frac{2h}{2}$
$\Rightarrow m+2m=-h$
$\Rightarrow h = − 3m \ldots\left(i\right)$
and $m_{1}m_{2}=\frac{a}{b}$
$\Rightarrow m\cdot2m=\frac{1}{2}$
$\Rightarrow m=\pm \frac{1}{2}\ldots\left(ii\right)$
From Eqs. $(i)$ and $ (ii)$, we get
$h=\pm\frac{3}{2}$