How would one approach proving the following statements:

(a) A plane can be covered with the interiors of finite number of hyperbolas

(b) A plane cannot be covered with the interiors of finite number of parabolas

Remarks:

interior is the part of the plane that contains a focus (foci) of the aforementioned shapes.

It is clear that without losing generality one can take $y=ax^2$ and $xy=a$ for all meaningful values of the parameter $a$ (e.g. $a\ne0$) and all possible rotations of these curves around the origin.