Official solution
Distance between foci = 2ae = 6
\[ \Rightarrow \] ae = 3 .....(1)
Distance between directrices = \[{{2a} \over e}\] = 12
\[ \Rightarrow \] \[{a \over e}\] = 6 .....(2)
from (1) and (2)
a2 = 18
also a2e2 = 9
\[ \Rightarrow \] 18e2 = 9
\[ \Rightarrow \] e2 = \[{1 \over 2}\]
We know e2 = 1 - \[{{{b^2}} \over {{a^2}}}\]
\[ \therefore \] \[{1 \over 2}\] = 1 - \[{{{b^2}} \over {{a^2}}}\]
\[ \Rightarrow \] b2 = 9
\[ \therefore \] Length of latus rectum = \[{{2{b^2}} \over a}\]
= \[{{2 \times 9} \over {\sqrt {18} }}\]
= \[3\sqrt 2 \]