$2x^2-13xy-7y^2+x+23y-6=0$

Point of intersection

$\frac{\partial f}{\partial x}=4x-13y+1=0$

$\frac{\partial f}{\partial y}=-14y-13x+23=0$

$\Rightarrow 4x\left(13x+14y-23\right)=0$

$\Rightarrow 13x(4x-13y+1)=0$

$\Rightarrow 225y-105=0$

$y=\frac{7}{15}$

$4x=\frac{91}{15}-1$

$x=\frac{19}{15}$

$P(\frac{19}{15},\frac{7}{15})$

$\partial =-\left(\frac{0+0-1}{{\displaystyle \frac{19}{15}}+{\displaystyle \frac{7}{15}}-1}\right)$

$\partial =\frac{15}{11}$