Pair of perpendicular lines passing through origin is

$a{x}^2+2hxy-a{y}^2=0$

or

$x^2+cxy-y^2=0$,

where $c=\frac{2h}{a}$

$\tan 45°=\left|\frac{m-\left({\displaystyle \frac{-2}{3}}\right)}{1+m\left({\displaystyle \frac{-2}{3}}\right)}\right|=1$

$\Rightarrow 3m+2=\pm \left(3-2m\right)$

$\Rightarrow m=-5,\frac{1}{5}$

Pair of perpendicular are, $\left(y+5x\right)\left(y-\frac{x}{5}\right)=0$

$\Rightarrow x^2-y^2+\left(\frac{-24}{5}\right)xy=0$, $c=\frac{-24}{5}$

$\Rightarrow 5\left(x^2-y^2\right)-24xy=0$