If the tangent to the curve $x = at^2, y = 2at$ is perpendicular to the $x$-axis,then what is the point of contact?

The equation of the normal to the parabola $y^{2} = x + a$ with slope $m$ is .....

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2$,then the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin,respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$,then the area (in $sq. \text{ units}$) of $\triangle SOR$ is equal to:

The vertex and the focus of the parabola $2y^2 + 5x - 6y + 1 = 0$ are respectively

Variable straight lines $y=mx+c$ make intercepts on the curve $y^2-4ax=0$ which subtend a right angle at the origin. Then the point of concurrence of these lines $y=mx+c$ is