Official solution
Focus (0, be) = (0, 5\[\sqrt 3\])
\[ \therefore \] be = 5\[\sqrt 3\]
\[ \Rightarrow \] b2e2 = 75
As here b > a
so e2 = \[1 - {{{a^2}} \over {{b^2}}}\]
\[ \therefore \] b2\[\left( {1 - {{{a^2}} \over {{b^2}}}} \right)\] = 75
\[ \Rightarrow \] b2 - a2 = 75 .......(1)
Given that,
difference of the lengths of major axis and minor
axis is 10.
\[ \therefore \] 2b - 2a = 10
\[ \Rightarrow \] b - a = 5 .......(2)
From (1),
(b + a)(b - a) = 75
\[ \Rightarrow \] (b + a)5 = 75
\[ \Rightarrow \] (b + a) = 15 .......(3)
From (2) and (3) we get,
b = 10, a = 5
\[ \therefore \] Length of its latus rectum = \[{{{2{a^2}} \over b}}\]
= \[{{2 \times 25} \over {10}}\] = 5