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 $(0,5\sqrt{3})$, then the length of its latus rectum is :

• A. $8$
• B. $10$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1(a>b)$
Given, $2a-2b=10\Rightarrow a-b=5\cdots \left(1\right)$
$c=5\sqrt{3}$
$\Rightarrow c^2=a^2-b^2=75$
$\Rightarrow (a+b)(a-b)=75$
$\Rightarrow a+b=15\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$a=10,b=5$
$\therefore$ Length of latus rectum $=\frac{2b^2}{a}=5$