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 (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

Mathematics · Ellipse

JEE Advanced 2021 — Paper 2 — Question 24

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 $\mathrm{Q}^{\prime}$ . Then the maximum possible value of the distance between $\mathrm{M}(\mathrm{P}, \mathrm{Q})$ and $\mathrm{M}\left(\mathrm{P}, \mathrm{Q}^{\prime}\right)$ , as $\mathrm{P}, \mathrm{Q}$ and $\mathrm{Q}^{\prime}$ vary on E , is $\qquad$

Answer: 4

Numerical answer — enter this value.

Step-by-step solution

$\mathrm{M}(\mathrm{P}, \mathrm{Q})$ is mid-point of PQ

$\mathrm{M}\left(\mathrm{P}, \mathrm{Q}^{\prime}\right)$ is mid-point of $\mathrm{PQ}^{\prime}$

In a $\triangle \mathrm{PQQ}^{\prime}$ since $\mathrm{M}(\mathrm{P}, \mathrm{Q}) \& \mathrm{M}\left(\mathrm{P}, \mathrm{Q}^{\prime}\right)$ are mid-point

of $\mathrm{PQ} \& \mathrm{PQ}^{\prime}$ hence line joining

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

is parallel to $\mathrm{QQ}^{\prime}$ and half of it. $\Rightarrow \max$

distance $=\frac{1}{2} \mathrm{QQ}^{\prime}=\frac{1}{2}(8)=4$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2021
Paper: Paper 2
Subject: Mathematics
Chapter: Ellipse
Topic: position of a Line or point wrt an Ellipse