JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Ellipse

The foci $(2,5)$ and $(2,-3)$ have the same $x$-coordinate, so the major axis is vertical with centre $(2,1)$, and the distance between them is $8$, giving $c=4$. With eccentricity $\frac{4}{5}$, the semi-major axis is $a=\frac{c}{e}=5$, and the semi-minor axis satisfies $b^2=a^2-c^2=25-16=9$. The latus rectum has length $\frac{2b^2}{a}=\frac{2(9)}{5}=\frac{18}{5}$.