Official solution
Let \[P(2\cos \theta ,\,\sqrt 2 \sin \theta )\] be any point on ellipse \[{{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\] and Q(4, 3) and let (h, k) be the mid point of PQ
then \[h = {{2\cos \theta + 4} \over 2},\,k = {{\sqrt 2 \sin \theta + 3} \over 2}\]
\[\therefore\] \[\cos \theta = h - 2,\,\sin \theta = {{2k - 3} \over {\sqrt 2 }}\]
\[\therefore\] \[{(h - 2)^2} + {\left( {{{2k - 3} \over {\sqrt 2 }}} \right)^2} = 1\]
\[ \Rightarrow {{{{(x - 2)}^2}} \over 1} + {{{{\left( {y - {3 \over 2}} \right)}^2}} \over {{1 \over 2}}} = 1\]
\[\therefore\] \[e = \sqrt {1 - {1 \over 2}} = {1 \over {\sqrt 2 }}\]