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 $$h = {{2\cos \theta + 4} \over 2},\,k = {{\sqrt 2 \sin \theta + 3} \over 2}$$

$$\therefore$$ $$\cos \theta = h - 2,\,\sin \theta = {{2k - 3} \over {\sqrt 2 }}$$

$$\therefore$$ $${(h - 2)^2} + {\left( {{{2k - 3} \over {\sqrt 2 }}} \right)^2} = 1$$

$$ \Rightarrow {{{{(x - 2)}^2}} \over 1} + {{{{\left( {y - {3 \over 2}} \right)}^2}} \over {{1 \over 2}}} = 1$$

$$\therefore$$ $$e = \sqrt {1 - {1 \over 2}} = {1 \over {\sqrt 2 }}$$