Question

If the line $x-2y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $(3,\frac{-9}{2})$, then the length of the latus rectum of the ellipse is:

• A. $9$
• B. $8\sqrt{3}$
• C. $12\sqrt{2}$
• D. $5$

Answer 1

Answer: (A)

$9$

Tangent at $(3,-\frac{9}{2})$
$\frac{3x}{a^2}-\frac{9y}{2b^2}=1$
Comparing this with $x-2y=12$
$\frac{3}{a^2}=\frac{9}{4b^2}=\frac{1}{12}$
We get $a=6$ and $b=3\sqrt{3}$
$L\left(LR\right)=\frac{2b^2}{a}=9$