Step-by-Step Solution

1. Analyze the Ellipse Equation The equation of the ellipse E is given as: ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$ This is a standard horizontal ellipse centered at the origin (0, 0). Comparing it with the standard form ${{{x^2}} \over {a^2}} + {{{y^2}} \over {b^2}} = 1$, we can identify the parameters:

   • $a^2 = 16 \implies a = 4$ (semi-major axis length)
   • $b^2 = 9 \implies b = 3$ (semi-minor axis length) The length of the major axis is $2a = 2(4) = 8$. The vertices are at $(\pm 4, 0)$.

2. Define the Points and Mid-points Let P, Q, and Q' be three distinct points on the ellipse E. Let their coordinates be:

   • $P = (x_P, y_P)$
   • $Q = (x_Q, y_Q)$
   • $Q' = (x_{Q'}, y_{Q'})$

   Let M(P, Q) be the mid-point of the line segment PQ, and M(P, Q') be the mid-point of the line segment PQ'. Using the midpoint formula, their coordinates are:

   • $M(P, Q) = \left( {{{x_P + x_Q}} \over 2}, {{{y_P + y_Q}} \over 2} \right)$
   • $M(P, Q') = \left( {{{x_P + x_{Q'}}} \over 2}, {{{y_P + y_{Q'}}} \over 2} \right)$

3. Calculate the Distance between the Mid-points Let $d$ be the distance between M(P, Q) and M(P, Q'). Using the distance formula: $d = \sqrt{ \left( {{{x_P + x_{Q'}}} \over 2} - {{{x_P + x_Q}} \over 2} \right)^2 + \left( {{{y_P + y_{Q'}}} \over 2} - {{{y_P + y_Q}} \over 2} \right)^2 }$ Simplifying the terms inside the square root: $d = \sqrt{ \left( {{{x_{Q'} - x_Q}} \over 2} \right)^2 + \left( {{{y_{Q'} - y_Q}} \over 2} \right)^2 }$ $d = \sqrt{ {1 \over 4} \left( (x_{Q'} - x_Q)^2 + (y_{Q'} - y_Q)^2 \right) }$ $d = {1 \over 2} \sqrt{ (x_{Q'} - x_Q)^2 + (y_{Q'} - y_Q)^2 }$ The expression inside the square root is the square of the distance between points Q and Q'. Let's denote this distance as $d(Q, Q')$. So, the relationship is: $d = {1 \over 2} d(Q, Q')$ This shows that the distance between the midpoints M(P, Q) and M(P, Q') is exactly half the distance between the points Q and Q'. Notice that the position of point P does not affect this distance.

4. Maximize the Distance To find the maximum possible value of $d$, we need to find the maximum possible value of the distance $d(Q, Q')$. Since Q and Q' are any two distinct points on the ellipse, we need to find the maximum distance between any two points on the ellipse.

   The maximum distance between any two points on an ellipse is the length of its major axis.

5. Calculate the Final Answer From Step 1, the length of the major axis of the ellipse is $2a = 8$. Therefore, the maximum value of $d(Q, Q')$ is 8. This occurs when Q and Q' are the vertices of the major axis, i.e., $Q=(4,0)$ and $Q'=(-4,0)$. The points P, Q, and Q' must be distinct. We can choose any other point for P, for example, $P=(0,3)$. This satisfies the condition of three distinct points.

   The maximum possible value of the distance $d$ is: $d_{max} = {1 \over 2} d(Q, Q')_{max} = {1 \over 2} \times 8 = 4$

Thus, the maximum possible value of the distance between M(P, Q) and M(P, Q') is 4.