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Suppose we have an (infinite) cone

\mathcal C

with apex

A

and a plane

\pi

. The intersection of

\pi

and

\mathcal C

is an ellipse

\mathcal E

with major axis

BC

, such that

B

is closer to

A

than

C

, and BC \equal{} 4, AC \equal{} 5, AB \equal{} 3. Suppose we inscribe a sphere in each part of

\mathcal C

cut up by

\mathcal E

with both spheres tangent to

\mathcal E

. What is the ratio of the radii of the spheres (smaller to larger)?

Problem Statement