Question

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $\left(0, 5\sqrt{3}\right),$ then the length of its latus rectum is:

• A. $ 10 $
• B. $ 8 $
• C. $ 5 $
• D. $ 6 $

Answer 1

Answer: (C)

Let equation of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
$2a - 2b = 10\quad\ldots\left(1\right)$
$ae = 5\sqrt{3}\quad\quad\ldots\left(2\right)$
$\frac{2b^{2}}{a} = ?$
$b^{2} = a^{2}\left(1-e^{2}\right)$
$b^{2} = a^{2}-a^{2}e^{2}$
$b^{2} = a^{2}-25 \times 3$
$\Rightarrow b = 5 $ and $ a = 10$
$\therefore $ length of L.R. $= \frac{2\left(25\right)}{10} = 5$