Mathematics - Parabola Question with Solution | TestHub

The parametric form $x=2t,y=\frac{t^2}{3}$ represents the parabola $x^2=12y$

Given $AS\perp AB$

So, $m_{AS}·m_{AB}=-1$

$\Rightarrow \frac{\left(3-\frac{t^2}{3}\right)}{\left(0-2t\right)}·\frac{\left(\alpha -\frac{t^2}{3}\right)}{\left(0-2t\right)}=-1$

 $\Rightarrow 3\alpha =\frac{27t^2+t^4}{t^2-9}$

Ordinate of centroid of $\Delta SAB=k=\frac{\alpha +\frac{t^2}{3}+3}{3}=\frac{9+3\alpha +t^2}{9}$

$\underset{t\to 1}{\lim }k=\underset{t\to 1}{\lim }\frac{1}{9}\left(9+t^2+\frac{27t^2+t^4}{\left(t^2-9\right)}\right)=\frac{13}{18}$