JEE MAIN 2026 Mathematics Conic Sections Ellipse Questions

Let an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is:

Let $A$ be the point $(3,0)$ and circles with variable diameter $AB$ touch the circle $x^2+y^2=36$ internally. Let the curve $C$ be the locus of the point $B$. If the eccentricity of $C$ is $e$, then $72e^2$ is equal to ____.