Cover a plane with parabolas and hyperbolas 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.
 A: For (a), the origin-centered rotations of $xy=1$ by $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$ leave a bounded octagonal hole that's easily covered.
For (b), consider a finite collection of parabolas, and choose a line $\ell$ parallel to none of their axes. (Since there are finitely-many axes, we can make such a choice.) The intersection of this $\ell$ with the interior any parabola, if not empty, is a finitely-long (possibly zero-length) segment. An infinitely-long line certainly contains a point not covered by a finite collection of those. 
A: Presumably you are allowed to have the interiors overlap.  For a, you need to find such a finite collection.  Think about $xy=1$ shifted up and right by some large amount, say $10$ units plus a copy shifted down and to the left by $10$ units.  The center gets covered nicely.  We still haven't got the whole plane, but ...
For b, think about trying to cover a very large circle.  To be specific, we will think about how much is covered by the interior of $y=x^2$.  If you take the circle $x^2+y^2=R^2$, you only cover from roughly $(-R,R^2)$ to $(R,R^2)$ (I know these aren't on the circle, but they almost are for large $R$ ).  This represents an angle of about $\frac 2R$ radians, which can be made arbitrarily small by taking $R$ large enough.  It works even if the parabola doesn't go through the origin, you just need to work on $R$.  So given a finite set of parabolas, take $R$ large enough that...
There are a lot of holes to fill, but the idea is there.
