2015 AMC 12A Problem 21

Problem 21 of 25HarderGeometry

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?

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Solution

The ellipse

\dfrac{x^2}{16} + y^2 = 1

has semi-axes

4

and

1,

c^2 = 16 - 1 = 15

and the foci are

(\pm\sqrt{15}, 0).

A circle through both foci has its center on the

y

-axis, say

(0, k),

with radius

\sqrt{k^2 + 15}.

Its top point always lies outside the ellipse. For four intersection points, its bottom point

(0, k - \sqrt{k^2 + 15})

must be below

y = -1,

which happens exactly when

0 \le k \lt 7.

k

ranges over

[0, 7),

the radius

\sqrt{k^2 + 15}

ranges over

[\sqrt{15}, 8),

a + b = \sqrt{15} + 8.

Thus, the correct answer is D.