Let A(α, 0) and B(0, β) be the points on the line 5 x+7 y=50. Let the point P divide the line segment A B internally in the ratio 7: 3. Let 3 x-25=0…

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