The combined equation of the angle bisectors of the lines represented by $2x^2+xy-y^2-10x+2y+8=0$ is

$a=2,2h=1,b=-1,2g=-10,2f=2,c=8$

Point of intersection is $(x_1,y_1)=\left(\frac{{\displaystyle hf-bg}}{{\displaystyle ab-h^2}},\frac{{\displaystyle gh-af}}{{\displaystyle ab-h^2}}\right)=(2,2)$

The pair of angular bisectors is $h\left({(x-x_1)}^2-{(y-y_1)}^2\right)=(a-b)(x-x_1)(y-y_1)$

$\Rightarrow \frac{1}{2}\left({(x-2)}^2-{(y-2)}^2\right)=(2+1)(x-2)(y-2)$

$\Rightarrow x^2-y^2-4x+4y=6(xy-2x-2y+4)$

$\Rightarrow x^2-6xy-y^2+8x+16y-24=0$