This month we have two related questions:

- For which values of
*a*are the parabolas*y*=*x*^{2}+*a*and

*x*=*y*^{2}+*a*tangent?

- For one value of
*a*, the interiors of the parabolas will be disjoint (left figure below), while for the other value the interiors will overlap (right figure below). Find the area of the region of overlap.