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