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21

Part of 2015 AMC 12/AHSME

Problems(1)

Circle Intersects an Ellipse

Source: 2015 AMC 12A #21

2/4/2015
A circle of radius rr passes through both foci of, and exactly four points on, the ellipse with equation x2+16y2=16x^2+16y^2=16. The set of all possible values of rr is an interval [a,b)[a,b). What is a+ba+b?
<spanclass=′latex−bold′>(A)</span>52+4<spanclass=′latex−bold′>(B)</span>17+7<spanclass=′latex−bold′>(C)</span>62+3<spanclass=′latex−bold′>(D)</span>15+8<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span>5\sqrt2+4\qquad<span class='latex-bold'>(B) </span>\sqrt{17}+7\qquad<span class='latex-bold'>(C) </span>6\sqrt2+3\qquad<span class='latex-bold'>(D) </span>\sqrt{15}+8\qquad<span class='latex-bold'>(E) </span>12
conicsellipsegeometrycircumcircleAMC