If the line $x + my + am^2 = 0$ is tangent to the parabola $y^2 = 4ax$,find the point of contact.

If the normal to the parabola ${y^2} = 12x$ at the point $(3, 6)$ meets the parabola again at the point $(27, -18)$,then the equation of the circle having this normal chord as its diameter is:

The equation of a tangent to the parabola,$x^2 = 8y,$ which makes an angle $\theta$ with the positive direction of the $x-$axis,is

For the parabola $y = \frac{h^3}{3} x^2 + \frac{h^2}{2} x - h + \frac{3}{4 h^3}$,if the equation of the directrix is $y = k$,then find the ratio $k : h$.

Let a circle tangent to the directrix of a parabola $y^2 = 2ax$ have its center coinciding with the focus of the parabola. Then the points of intersection of the parabola and the circle are

$P$ and $Q$ are two distinct points on the parabola,$y^2 = 4x$,with parameters $t$ and $t_1$ respectively. If the normal at $P$ passes through $Q$,then the minimum value of $t_1^2$ is