Question: Line $l_1:2x-3y+2=0$ and $l_2:3x-2y+3=0$ is the chord of circle with length $26$ and $24$ units respectively. Find the equation of circle.
Is the solution unique? If not unique, what is the general form of the solution?
My attempt so far: Let circle to have radius $r$ and center $C(h,k)$ and apply Pythagoras theorem. So I have two equations $\sqrt{r^2-13^2}=\frac{|2h-3k+2|}{\sqrt{2^2+(-3)^2}}$ and $\sqrt{r^2-12^2}=\frac{|3h-2k+3|}{\sqrt{3^2+(-2)^2}}$. Eliminating the $r$ and I obtain the locus of $C$ which is a hyperbola $(h+1)^2-k^2=65$.