Mathematics - Hyperbola Question with Solution | TestHub

Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive x -axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3x+2y=1$. Let H be the hyperbola $\frac{x^2}{{\alpha }^2}-\frac{y^2}{{\beta }^2}=1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2{\alpha }^2+3{\beta }^2$ is equal to ______