JEE Main 2024 — Ellipse Question with Solution

Let $$A(\alpha ,0)$$ and $$B(0,\beta )$$ be the points on the line $$5x+7y=50$$. Let the point $$P$$ divide the line segment $$AB$$ internally in the ratio $$7:3$$. Let $$3x-25=0$$ be a directrix of the ellipse $$E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ and the corresponding focus be $$S$$. If from $$S$$, the perpendicular on the $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to,