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

• A

  $6$
• B

  $10$
• C

  $8$
• Correct answer
  D

  $5$

Solution

Given, difference of the lengths of major and minor axis is $10$.
$\Rightarrow 2b-2a=10$ … (1)
One of the focii is $\left(0, 5\sqrt{3}\right)$
$\Rightarrow be=5\sqrt{3}$ … (2)
Also $a^2=b^2\left(1-e^2\right)$ .......(3)
$\Rightarrow a^2=b^2-75$
$\Rightarrow \left(b-a\right)\left(a+b\right)=75$
$\Rightarrow a+b=15$ … (4)
From (1) and (4)
$a=5$ and $b=10$
Hence length of the latus rectum $=\frac{2a^2}{b}=2\cdot \frac{25}{10}=5$

Hence, the correct answer is option: 'D'.