 Solution

The equation is hyperbolic at a point when $\boxed{b^2-ac>0}$. The principal form has one positive and one negative eigenvalue and two distinct real characteristic directions. In the given equation $a=x^2$, $b=0$, $c=-y^2$, so $b^2-ac=x^2{y}^2>0$ throughout the positive quadrant, consistent with the two canonical characteristic families used above.