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Question

Let E be the ellipse x216+y29=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

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Solution

Let M(P,Q)=M and M(P,Q)=M
By mid point theorem:
MM=12(QQ)
max(MM)=12×max(QQ)
Maximum of QQ is possible when QQ= major axis.
QQ=8
max(MM)=12×8=4

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