Let $A(\alpha,0)$ and $B(0,\beta)$ be 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 $E:\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and let the corresponding focus be $S$. If the perpendicular from $S$ to the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is: