Official solution
3x + 4y = 12\[\sqrt 2 \]
\[ \Rightarrow \] y = \[ - {{3x} \over 4} + 3\sqrt 2 \]
is tangent to
\[{{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1\]
\[ \therefore \] c2
= a2m2 + b2
\[ \Rightarrow \] \[{\left( {3\sqrt 2 } \right)^2} = {a^2}{\left( { - {3 \over 4}} \right)^2} + 9\]
\[ \Rightarrow \] a2 = 16
Also e = \[\sqrt {1 - {{{b^2}} \over {{a^2}}}} \]
= \[\sqrt {1 - {9 \over {16}}} \] = \[{{\sqrt 7 } \over 4}\]
Distance between focii = 2ae
= \[2 \times 4 \times {{\sqrt 7 } \over 4}\]
= \[2\sqrt 7 \]