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 $M \left( P , Q '\right)$ 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 $M \left( P , Q '\right)$, as $P , Q$ and $Q'$ vary on $E$, is _____

Answer 1

Let P(α), Q(θ), Q’(θ’)

M = ½ (4 cos α + 4 cos θ), ½ (3 sin α + 3 sin θ)

M’ = ½ (4 cos α + 4 cos θ’), ½ (3 sin α + 3 sin θ’)

MM’ = ½ √((4 cos θ – 4 cos θ’)2 + (3 sin θ – 3 sin θ’)2)

MM’ = ½ distance between Q and Q’

MM’ is not depending on P

Maximum of QQ’ is possible when QQ’ = major axis

QQ’ = 2(4) = 8

MM’ = ½ (QQ’)

MM’ = 4