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  4. Let A be a fixed point (0,6) and B be a moving point (2 t , 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

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Q. Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2 t , 0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is :

JEE MainJEE Main 2021Straight Lines

Solution:

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Perpendicular bisector of $AB$ is
$(y-3)=\frac{t}{3}(x-t)$
So, $C=\left(0,3-\frac{t^{2}}{3}\right)$
Let $P$ be $( h , k )$
$h =\frac{ t }{2} ; k =\left(3-\frac{ t ^{2}}{6}\right)$
$\Rightarrow k =3-\frac{4 h ^{2}}{6} $
$\Rightarrow 2 x ^{2}+3 y -9=0$