A calculus problem by Hobart Pao

Given the parabolas $y = x^2, y=-x^2, x=y^2, x=-y^2$ , an osculating circle can be drawn on each parabola at the origin.

You now have 4 osculating circles, and they create four areas of intersection.

The total area of these 4 intersections is equal to the area of intersection of the two circles $r= a \cos \theta$ and $r=a \sin \theta$ . Find $a$ .

Clarification:

The osculating circle to $\vec{r}(t)$ at time $t$ is the circle tangent to $\vec{r}(t)$ at time $t$ with radius $\dfrac{1}{\kappa (t)}$ , where $\kappa(t)$ is the curvature of $\vec{r}(t)$ at time $t$ .