Question

Let $E$ be the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$. For any three distinct points $P, Q$ and $Q'$ on $E$, let $M(P, Q)$ be the midpoint of the line segment joining $P$ and $Q$, and $M \left( P , Q '\right)$ be the mid-point of the line segment joining $P$ and $Q'$. Then the maximum possible value of the distance between $M ( P , Q )$ and $M \left( P , Q '\right)$, as $P , Q$ and $Q'$ vary on $E$, is _____

Answer 1

$M ( P , Q )$ is mid-point of $PQ$
$M \left( P , Q'\right)$ is mid-point of $PQ ^{\prime}$
In a $\triangle PQQ'$ since $M ( P , Q ) \& M \left( P , Q '\right)$ are mid-point
of $PQ \& PQ'$ hence line joining M(P,Q), M(P,Q') is
parallel to $QQ'$ and half of it.
$\Rightarrow \max$ distance $=\frac{1}{2} Q Q'=\frac{1}{2}(8)=4$