# A calculus problem by Hobart Pao

Calculus Level 4

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 $$t$$.

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