In an ellipse, with centre at the origin,


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. (1) 10

  2. (2) 5

  3. (3) 8

  4. (4) 6

Correct Option: , 2


Given that focus is $(0,5 \sqrt{3}) \Rightarrow|b|>|a|$

Let $b>a>0$ and foci is $(0, \pm b e)$

$\because a^{2}=b^{2}-b^{2} e^{2} \Rightarrow b^{2} e^{2}=b^{2}-a^{2}$

$b e=\sqrt{b^{2}-a^{2}} \Rightarrow b^{2}-a^{2}=75$.....(i)

$\because 2 b-2 a=10 \Rightarrow b-a=5 \quad \ldots$ (ii)

From (i) and (ii)


On solving (ii) and (iii), we get

$\Rightarrow b=10, a=5$

Now, length of latus rectum $=\frac{2 a^{2}}{b}=\frac{50}{10}=5$

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