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

Question

Let $A(\alpha,0)$ and $B(0,\beta)$ be 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 an ellipse $E$, and the corresponding focus be $S$. If the perpendicular from $S$ to the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to:

Accepted Answer

$A=(10,0),\ B=\left(0,\dfrac{50}{7}\right)$. Dividing internally $7:3$ (from $A$ to $B$):

$$P=\left(\frac{3(10)}{10},\frac{7(50/7)}{10}\right)=(3,5).$$

$S$ has the same $x$-coordinate as $P$: $S=(3,0)$, so $ae=3$. Directrix: $\dfrac{a}{e}=\dfrac{25}{3}$.

$$a^2=3\times\frac{25}{3}=25\Rightarrow a=5,\ e=\frac35,\qquad b^2=25\left(1-\frac9{25}\right)=16.$$

$$\text{Latus rectum}=\frac{2b^2}{a}=\frac{32}{5}.$$

Solutions · 1