If l x 2 + 3 x y - 2 y 2 - 5 x + 5 y + k = 0 represents a pair of perpendicular lines, then

$l{x}^2+3xy-2y^2-5x+5y+k=0$

if lines are perpendicular

then, $l-2=0$

$\Rightarrow l=2$

$2x^2+3xy-2y^2-5x+5y+k=0$

it represents a pair of lines

$a=2, b=-2, c=k$

$h=\frac{3}{2}, f=\frac{5}{2}, g=\frac{-5}{2}$

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\Rightarrow -4k-\frac{75}{4}-\frac{50}{4}+\frac{2x25}{4}-\frac{9k}{4}=0$

$\Rightarrow 25k=-75$

$\Rightarrow k=-3$