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