The locus of the point of intersection of the tangents drawn at the ends of a focal chord of the parabola $x^{2}=-8y$ is

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

What is the equation of the directrix of the parabola $x^2 = -8y$?

If ${x^2} + 6x + 20y - 51 = 0$,then the axis of the parabola is

Find the shortest distance of the point $(0, c)$ from the parabola $y=x^{2}$ where $0 \leq c \leq 5$.

$A$ normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$,where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$,then the value of $24a$ is. . . .