Let ABCD be a quadrilateral. If E and F are the mid points of the...

Given,

$ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and \left(\overset{\to}{AB} - \overset{\to}{BC}\right) + \left(\overset{\to}{AD} - \overset{\to}{DC}\right) = k\overset{\to}{FE}$,</p><p>Now plotting the diagram we get,</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/eca13767-e7f3-495a-ac75-da94a4643863.png" style="width: 280px; height: 254px;" /></p><p>Now let position vector of$A, B, C$and$D$are$\overset{\to}{a,}\overset{\to}{b,}\overset{\to}{c}$and$\overset{\to}{d}$respectively</p><p>Now by midpoint formula,</p><p>Position vector of$E = \dfrac{\overset{\to}{OC} + \overset{\to}{OA}}{2} = \dfrac{\overset{\to}{c} + \overset{\to}{a}}{2}$</p><p>And Position vector of$F = \dfrac{\overset{\to}{b} + \overset{\to}{d}}{2}$</p><p>Now, \left(\overset{\to}{AB} - \overset{\to}{BC}\right) + \overset{\to}{AD} - \overset{\to}{DC}$

$⇒ \overset{\to}{b} - \overset{\to}{a} - \left(\overset{\to}{c} - \overset{\to}{b}\right) + \overset{\to}{d} - \overset{\to}{a} - \left(\overset{\to}{c} - \overset{\to}{d}\right)$

$⇒ 2\overset{\to}{b} - 2\overset{\to}{a} - 2\overset{\to}{c} + 2\overset{\to}{d}$

$⇒ 2\left(\overset{\to}{b} + \overset{\to}{d}\right) - 2\left(\overset{\to}{a} + \overset{\to}{c}\right)$

$⇒ 4\left[\dfrac{\overset{\to}{b} + \overset{\to}{d}}{2} - \dfrac{\overset{\to}{a} + \overset{\to}{c}}{2}\right] = 4\left[\overset{\to}{OF} - \overset{\to}{OE}\right]$

$⇒ 4\overset{\to}{EF} = - 4\overset{\to}{FE}$

$∴ k = - 4$