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

• A

  $\frac{25}{3}$
• B

  $\frac{32}{9}$
• C

  $\frac{25}{9}$
• Correct answer
  D

  $\frac{32}{5}$

Hence, the correct answer is option: 'D'.