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 mid-point of the line segment joining $P$ and $Q$, and $M(P,Q^{\prime })$ 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 $M(P,Q^{\prime }),$ as $P,Q$ and $Q^{\prime }$ vary on $E$, is

Answer 1

Let $M(P,Q)=M$ and $M(P,Q^{\prime })=M^{\prime }$
By mid point theorem:
$M{M}^{\prime }=\frac{1}{2}\left(Q{Q}^{\prime }\right)$
$\Rightarrow max\left(M{M}^{\prime }\right)=\frac{1}{2}\times max\left(Q{Q}^{\prime }\right)$
Maximum of $Q{Q}^{\prime }$ is possible when $Q{Q}^{\prime }=$ major axis.
$\therefore Q{Q}^{\prime }=8$
$\Rightarrow max\left(M{M}^{\prime }\right)=\frac{1}{2}\times 8=4$