2015 AMC 12A Problem 21

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

\textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.