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 3 , then the length of its latus rectum is:

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

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