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 $\left( 3, \frac{-9}{2} \right)$, then the length of the latus recturm of the ellipse is :

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

Answer 1

Answer: (A)

Tangent at $\left(3, - \frac{9}{2}\right) $
$ \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 $