JEE Advanced 2021 Paper 2 · Q18 · Ellipse

Question

Let $E$ be the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. For any three distinct points $P, Q$ and $Q'$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$, and $M(P, Q')$ be the mid-point 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 _____.

SolveKar AI · Accepted Answer

$M(P,Q) - M(P,Q') = \dfrac{P+Q}{2} - \dfrac{P+Q'}{2} = \dfrac{Q - Q'}{2}$. So the distance between the two midpoints equals half the distance between $Q$ and $Q'$. The maximum distance between two points on the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$ is the major axis length $= 2a = 8$. Hence the maximum is $\dfrac{8}{2} = 4$.

JEE Advanced 2021 Paper 2 · Q18 · Ellipse

Solution