Ellipse Question 5
Question 5 - 2024 (30 Jan Shift 2)
Let $\mathrm{A}(\alpha, 0)$ and $\mathrm{B}(0, \beta)$ 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 $3 x-25=0$ 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 $\mathrm{P}$, then the length of the latus rectum of $E$ is equal to
(1) $\frac{25}{3}$
(2) $\frac{32}{9}$
(3) $\frac{25}{9}$
(4) $\frac{32}{5}$
Show Answer
Answer (4)
Solution
$\left.\begin{array}{l}\mathrm{A}=(10,0) \ \mathrm{B}=\left(0, \frac{50}{7}\right)\end{array}\right} \mathrm{P}=(3,5)$
ae $=3$
$\frac{\mathrm{a}}{\mathrm{e}}=\frac{25}{3}$
$\mathrm{a}=5$
$\mathrm{b}=4$
Length of $L R=\frac{2 b^{2}}{a}=\frac{32}{5}$
