Let x=2t, y=t^23 be a conic.

Question

Let $x=2t,\ y=\dfrac{t^2}3$ be a conic. Let $S$ be the focus, and $B$ be the point on the axis of the conic such that $SA\perp BA$, where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of $	riangle SAB$, then $\displaystyle\lim_{t	o1}k$ is equal to:

Accepted Answer

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

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

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

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

Given $$SA\, \bot \,BA$$

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

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

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

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

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

Solutions · 1