MathDB

An ellipse has foci

A

and

B

and has the property that there is some point

C

on the ellipse such that the area of the circle passing through

A

B

, and,

C

is equal to the area of the ellipse. Let

e

be the largest possible eccentricity of the ellipse. One may write

e^2

\frac{a+\sqrt{b}}{c}

, where

a, b

, and

c

are integers such that

a

and

c

are relatively prime, and b is not divisible by the square of any prime. Find

a^2 + b^2 + c^2

Problem Statement