Let [A(, 0)] and [B(0, )] be the points on the line 5 x+7 y=50.

Question

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,

Accepted Answer

$$\left.\[\begin{gathered} \mathrm{A}=(10,0) \ \mathrm{B}=\left(0, \frac{50}{7}\right) \end{gathered}$$\right\} \mathrm{P}=(3,5)\]

$$\begin{aligned}
& \text { ae }=3 \
& \frac{\mathrm{a}}{\mathrm{e}}=\frac{25}{3} \
& \mathrm{a}=5 \
& \mathrm{~b}=4
\end{aligned}$$

Length of $$L R=\frac{2 b^2}{a}=\frac{32}{5}$$

Solutions · 1