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 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 ___.

Let E be the ellipse x²/16 + y²/9 = 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 ___. $E: \dfrac{x^2}{16} + \dfrac{y^2}{9} = 1 \Rightarrow a = 4,\ b = 3$ $M(P,Q),\ M(P,Q') \text{ are mid-points of } PQ \text{ and } PQ'$ $\text{To find: } \max\ \big|\ M(P,Q) - M(P,Q')\ \big|$ $E: a^2 = 16,\ b^2 = 9 \Rightarrow \text{major axis } = 2a = 8$ $1.\ M(P,Q),\ M(P,Q')$ $2.\ M(P,Q) - M(P,Q') = ?$ $3.\ \max |QQ'| = 2a$ $M(P,Q) = \dfrac{P+Q}{2}$ $M(P,Q') = \dfrac{P+Q'}{2}$ $M(P,Q) - M(P,Q') = \dfrac{P+Q}{2} - \dfrac{P+Q'}{2} = \dfrac{Q-Q'}{2}$ $\big|M(P,Q) - M(P,Q')\big| = \dfrac{|QQ'|}{2}$ $M_1M_2 \parallel QQ',\quad M_1M_2 = \dfrac{QQ'}{2}\ \text{(mid-segment)}$ $|QQ'| \le 2a = 8 \Rightarrow M_1M_2 \le \dfrac{8}{2} = 4$ $P(0,3),\ Q(-4,0),\ Q'(4,0) \Rightarrow M_1(-2,\tfrac32),\ M_2(2,\tfrac32)$ $M_1M_2 = 4 = \dfrac{QQ'}{2}$ $\boxed{4}$