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

Question

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,

Accepted Answer

$\begin{gathered} A=(10,0) \ B=(0,\frac{50}{7}) \end{gathered}P=(3,5)$

$\begin{gathered} =3 \ \frac{a}{e}=\frac{25}{3} \ a=5 \ b=4 \end{gathered}$

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

Solutions · 1