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

Let the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Foci $=(0,5\sqrt{3})\Rightarrow$ major axis = 2b, minor axis = 2a, be $=5\sqrt{3}$

Given $2b-2a=10\Rightarrow b=a+5$

be $=5\sqrt{3}\Rightarrow b^2{e}^2=75\Rightarrow b^2-a^2=75\Rightarrow (a+5{)}^2-a^2=75\Rightarrow 10a=50\Rightarrow a=5,b=10$

Length of latus rectum $=\frac{2a^2}{b}=\frac{2(5{)}^2}{10}=5$