Highly Elliptical Area

Geometry Level 5

An ellipse is inscribed in a right triangle such that its major and minor axes are parallel to the perpendicular legs of the right triangle.

If the area of the ellipse is A1A_1, the area of the right triangle is A2A_2, and the maximum value of A1A2\dfrac {A_1}{A_2} is π(ab)\pi (a - \sqrt b), where aa and bb are positive integers, find a2+b2a^2+b^2.

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