VWO 1997 Problem 2

In the cartesian plane, consider the curves $x^2+y^2=r^2$ and $(xy)^2=1$ . Call $F_r$ the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)

(a) Find the area of the polygon as a function of $r$ . (b) For which values of $r$ do we have a regular polygon?