JEE MAIN - Mathematics (2024 - 30th January Evening Shift - No. 20)
Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ 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 $$P$$, then the length of the latus rectum of $$E$$ is equal to,
$$\frac{25}{3}$$
$$\frac{25}{9}$$
$$\frac{32}{5}$$
$$\frac{32}{9}$$
Explanation
$$\left.\begin{array}{l} \mathrm{A}=(10,0) \\ \mathrm{B}=\left(0, \frac{50}{7}\right) \end{array}\right\} \mathrm{P}=(3,5)$$
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$$\begin{aligned} & \text { ae }=3 \\ & \frac{\mathrm{a}}{\mathrm{e}}=\frac{25}{3} \\ & \mathrm{a}=5 \\ & \mathrm{~b}=4 \end{aligned}$$
Length of $$L R=\frac{2 b^2}{a}=\frac{32}{5}$$
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