2025 AMC 12A 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 = \dfrac{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?

\textbf{(A)} \dfrac{3}{5} \qquad \textbf{(B)} \dfrac{16}{25} \qquad \textbf{(C)} \dfrac{4}{5} \qquad \textbf{(D)} \dfrac{22}{23} \qquad \textbf{(E)} \dfrac{44}{45}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.