Ellipse
Distance Between Points
JEE Advanced 2021 Paper 2
Grade 11
Question:
Let $E$ be the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$. For any three distinct points $P, Q, Q'$ on $E$, let $M(P, Q)$ be the midpoint of $PQ$ and $M(P, Q')$ be the midpoint of $PQ'$. Then the maximum possible distance between $M(P, Q)$ and $M(P, Q')$ as $P, Q, Q'$ vary on $E$ is _____.
Step-by-Step Solution
Key Concept: $M(P, Q) = \frac{P + Q}{2}$ and $M(P, Q') = \frac{P + Q'}{2}$. Their distance $= \frac{|QQ'|}{2}$. This is maximised when $|QQ'|$ is maximised, i.e., $Q$ and $Q'$ are the farthest points on $E$. The major axis has length $2a = 8$, so max distance $= 4$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 4