It is easy to see that the epigraph of a parabola, i.e. the set $ \\{(x,y)\in \mathbb R^2, y> x^2\\} $ is a countable increasing union of ellipses in the sense that $$ \\{(x,y)\in \mathbb R^2, y> x^2\\}=\cup_{k\ge 1}\\{(x,y)\in \mathbb R^2, y\ge x^2+k^{-2} y^2\\}. $$ On the other hand, I believe that the epigraph of an hyperbola defined as $$\\{(x,y)\in \mathbb R^2, y> \sqrt{x^2+1}\\}$$ cannot be a countable increasing union of ellipses. However, the proof that I have of this fact is very indirect and is using some rather complicated stuff about singular integrals. It is quite likely that there is a simple planar Euclidean geometry argument to support the above claim. Maybe a simple argument about the eccentricity of the hyperbola (which is $>1$) can prevent that it is the union of ellipses (whose eccentricity is $<1$).