QJEEM22Mafd35: JEE Main 2022 Mathematics - Class 11 JEEM22Mafd35 Solution: Let , be a conic. Let S be the focus and B be the point on the…

$x = 2t,\,y = {2 \over 3}$

$t \to 1\,\,\,A \equiv \left( {2,{1 \over 3}} \right)$

Given conic is ${x^2} = 12y \Rightarrow S \equiv (0,3)$

Let $B \equiv (0,\beta )$

Given $SA\, \bot \,BA$

$\left( {{{{1 \over 3}} \over {2 - 3}}} \right)\left( {{{\beta - {1 \over 3}} \over { - 2}}} \right) = - 1$

$\Rightarrow \left( {\beta - {1 \over 3}} \right){1 \over 3} = - 2$

$\Rightarrow \beta = {1 \over 3}\left( {{{ - 17} \over 3}} \right)$

$\mathop {Ordinate\,of\,centroid}\limits_{(as\,t \to 1)} = k = {{\beta + {1 \over 3} + 3} \over 3}$

$= {{{{ - 17} \over 9} + {{10} \over 3}} \over 3} = {{13} \over {18}}$