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

Correct Answer: 4

Step 1: Analyzing the Given Ellipse Equation
We are given the ellipse equation: $$ \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1. $$
This represents an ellipse with semi-major axis $a = 4$ (along the x-axis) and semi-minor axis $b = 3$ (along the y-axis).
Step 2: Points on the Ellipse
We are given three points $P = (x_1, y_1)$, $Q = (x_2, y_2)$, and $Q' = (x_3, y_3)$, each satisfying the ellipse equation: $$ \frac{x_1^{2}}{16} + \frac{y_1^{2}}{9} = 1, \quad \frac{x_2^{2}}{16} + \frac{y_2^{2}}{9} = 1, \quad \frac{x_3^{2}}{16} + \frac{y_3^{2}}{9} = 1. $$
These points lie on the ellipse, which we will use to find the midpoints.
Step 3: Coordinates of the Midpoints
The coordinates of the midpoint $M(P, Q)$ of the segment connecting $P$ and $Q$ are: $$ M(P, Q) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right). $$
Similarly, the coordinates of the midpoint $M(P, Q')$ of the segment connecting $P$ and $Q'$ are: $$ M(P, Q') = \left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}\right). $$
Step 4: Distance Between the Two Midpoints
The distance $d$ between the two midpoints $M(P, Q)$ and $M(P, Q')$ is given by the Euclidean distance formula: $$ d = \sqrt{\left(\frac{x_2 - x_3}{2}\right)^2 + \left(\frac{y_2 - y_3}{2}\right)^2} = \frac{1}{2} \sqrt{(x_2 - x_3)^2 + (y_2 - y_3)^2}. $$
Step 5: Maximizing the Distance
To maximize the distance between the two midpoints, we need to maximize the Euclidean distance between the points $Q(x_2, y_2)$ and $Q'(x_3, y_3)$. This is equivalent to maximizing: $$ \sqrt{(x_2 - x_3)^2 + (y_2 - y_3)^2}. $$
The maximum distance between two points on the ellipse occurs when they are located at the two farthest points along the major axis of the ellipse.
Step 6: Farthest Points on the Ellipse
The major axis of the ellipse is along the x-axis, and the maximum distance between two points on the ellipse occurs when they are at the endpoints of the major axis, which are located at $x = 4$ and $x = -4$ (since the semi-major axis is $a = 4$). Therefore, the maximum distance between $Q$ and $Q'$ is: $$ \text{Distance between } Q \text{ and } Q' = 4 - (-4) = 8. $$
Step 7: Maximum Possible Value of the Distance Between the Midpoints
Now, the maximum possible value of the distance $d$ between the midpoints is: $$ d = \frac{1}{2} \times 8 = 4. $$
Therefore, the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q')$ is $4$.
Thus, the solution is confirmed: the maximum distance is $\boxed{4}$.