Question

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

Answer 1

Answer: 4

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