QJEEM22Ma06b3: JEE Main 2022 Mathematics - Class 11 JEEM22Ma06b3 Solution: The locus of the mid point of the line segment joining the…

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

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

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

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

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