"Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :"

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Accepted Answer

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

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

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

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

Given $SA\, \bot \,BA$

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

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

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

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

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

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Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

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Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

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