In an ellipse, with centre at the origin,

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:

  1. 10

  2. 8

  3. 5

  4. 6


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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