If focus of the parabola which touches $x=0,y=0,x-y+1=0$ and $-2x-y-8=0$ is $(\alpha ,\beta )$, then $\left|\frac{\alpha }{\beta }\right|$ is_____

Circle passing through points of intersection of tangents also passes through focus. Equation of two circles is $x^2+y^2+x-y=0, x^2+y^2+4x+8y=0$. Equation of common chord is $x+3y=0.S(\alpha ,\beta )\text{ lies on it}$