If I consider the intersection of the plane $x=0$ and the elliptic hyperboliede of a sheet $\frac{x^{2}}{4}+\frac{y^{2}}{4}-\frac{z^{2}}{9}=1$, we obtain that the intersection is the hyperbola in the yz plane
$$\frac{y^{2}}{4}-\frac{z^{2}}{9}=1$$
and its parameterization would be
$$X(t)=(0,2\sec(t),3\tan(t))$$
I try to find two different quadratic surfaces of elliptic planes and hyperbolas of a sheet that when intersected also obtain a hyperbola.