Let $E$ be the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$. For any three distinct points $P, Q$ an

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 $M\left(P, Q^{\prime}\right)$ 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

Professor PC Thomas Classes · Answer

(4) :\ Let $M_{1} ightarrow$ midpoint of $P Q$\ $M_{2} ightarrow$ midpoint of $P Q^{\prime}$\ We have, $M_{1} M_{2}=\frac{1}{2} Q Q^{\prime}$. To get the maximum value of $M_{1} M_{2}$, we can choose $Q$, $Q^{\prime}$ to be the end of the major axis.\ Hence, $M_{1} M_{2 	ext { max. }}=\frac{1}{2} A A^{\prime}=\frac{1}{2} \cdot 8=4$\