AMC 12A 2025 (Problem 14)

Points

F

G

, and

H

are collinear with

G

between

F

and

H

. The ellipse with foci at

G

and

H

is internally tangent to the ellipse with foci at

F

and

G

, as shown below. The two ellipses have the same eccentricity

e

, and the ratio of their areas is

2025

. (Recall that the eccentricity of an ellipse is

e=\frac{c}{a}

, where

c

is the distance from the center to a focus, and

2a

is the length of the major axis.) What is

e

\text{(A)}\;\frac{3}{5}

\text{(B)}\;\frac{16}{25}

\text{(C)}\;\frac{4}{5}

\text{(D)}\;\frac{22}{23}

\text{(E)}\;\frac{44}{45}