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MathematicsEllipseChord with given Middle PointMedium2 minPYQ_2021

MathematicsMediumnumerical

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 _____.

Answer:

4.00

Solution:

In a triangle sides joining the mid-points of two sides is half and parallel to third side

$M_{1} M_{2} = \frac{1}{2} Q Q^{'}$

Maximum value of $Q Q^{'}$ is $A A^{'}$

Hence, maximum value of $M_{1} M_{2} = \frac{1}{2} A A^{'} = 4$

Stream:JEE_ADVSubject:MathematicsTopic:EllipseSubtopic:Chord with given Middle Point

⏱ 2mℹ️ Source: PYQ_2021

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