Ellipse and Circle Love

Geometry Level pending

A circle of radius r r passes through both foci of, and exactly four points on, the ellipse with equation x 2 + 16 y 2 = 16 x^2 + 16y^2 = 16 . The set of all possible values of r r is an interval [ a , b ) [a,b) . What is a + b a + b ?

This is a problem from the 2015 AMC (12 A #21).
1 3 1 2 + 9 13^\frac{1}{2} + 9 1 7 1 2 + 7 17^\frac{1}{2} + 7 1 2 1 2 + 8 12^\frac{1}{2} + 8 1 5 1 2 + 8 15^\frac{1}{2} + 8

A Former Brilliant Member by A Former Brilliant Member

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