Let be the ellipse For any three distinct points and on , let be the midpoint of the line segment joining and , and be the midpoint of the line segment joining and . Then the maximum possible value of the distance between and , as and vary on ,

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 $M (P, Q')$ be the midpoint of the line segment joining $P$ and $Q'$ . Then the maximum possible value of the distance between $M (P, Q)$ and $M (P, Q')$ , as $P, Q$ and $Q'$ vary on $E$ , is ______. [2021]

Smart Achievers · Accepted Answer

(4)

$Let A and B be midpoints of segment P Q and P Q' respectively$

$By midpoint theorem$

$Q'$

$∴$ $Distance between M (P, Q) and M (P, Q') = \frac{1}{2} Q Q'$

$Since, Q, Q' must be on E, so, maximum of Q Q' = 8$

$∴$ $Maximum of A B = \frac{8}{2} = 4$

Solution

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Solution

Ans. (4) Let A and B be midpoints of segment PQ and PQ' respectively By midpoint theorem AB∥QQ' and AB=12QQ' ∴ Distance between M(P,Q) and M(P,Q')=12QQ' Since, Q,Q' must be on E, so, maximum of QQ'=8 ∴ Maximum of AB=82=4

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