Let E be ellipse frac{x^2}{16}+frac{y^2}{9}=1 . For any three distinct points P, Q and Q^{prime} on

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10-2. Parabola, Ellipse, Hyperbola

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Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$, and $M \left( P , Q ^{\prime}\right)$ be the mid-point of the line segment joining $P$ and $Q ^{\prime}$. Then the maximum possible value of the distance between $M ( P , Q )$ and $M \left( P , Q ^{\prime}\right)$, as $P, Q$ and $Q^{\prime}$ vary on $E$, is. . . . .

$2$

$3$

$4$

$5$

(IIT-2021)

Solution

$A$ and $B$ be midpoints of segment $PQ$ and $PQ ^{\prime}$ respectively $AB =$ distance between $M ( P , Q )$ and $M \left( P , Q ^{\prime}\right)=\frac{1}{2} \cdot QQ ^{\prime}$

Since, $Q , Q$ ' must be on $E$, so, maximum of $QQ ^{\prime}=8$

$\therefore$ Maximum of $AB =\frac{8}{2}=4$

Standard 11

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