Partial Derivatives & the Gradient · Hard problem

Near most points, the equation $x^{2}+y^{2}-z^{2}=5$ defines $z$ as a smooth function of $x$ and $y$ — except where $\dfrac{\partial F}{\partial z}=0$ (so the Implicit Function Theorem's hypothesis fails), where $F=x^2+y^2-z^2-5$. At such a point, find $x^{2}+y^{2}$.