Answer
The intersection is empty if and only if $ h\in[-\frac{1}{2},\frac{1}{2}]$.
Work Step by Step
Putting $ z=h $ in the equation, we have
$$-x^2-4y^2+4h^2=1$$
hence we get
$$ x^2+4y^2=1-4h^2.$$
The intersection is non-empty whenever
$$4h^2-1\gt 0\Longrightarrow(h+\frac{1}{2})(h-\frac{1}{2})\gt 0$$
hence $$ h\in (-\infty,-\frac{1}{2})\cup (\frac{1}{2},\infty) .$$
So, the intersection is empty if and only if $ h\in[-\frac{1}{2},\frac{1}{2}]$.
