Tangents are drawn at three points $P(t_1), Q(t_2), R(t_3)$ on the parabola $y^2 = x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1 = 2, t_2 = -4, t_3 = 6$,then the area of the triangle $LMN$ is

$A$ 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. . . .

If a focal chord of the parabola $y^2 = 4ax$ makes an angle $\theta$ with its axis,then the length of the perpendicular from the vertex to this chord is......

If the vertex of a parabola is $(0, a)$ and the focus is $(0, 0)$,what is its equation?

Statement $1$: $y = mx - \frac{1}{m}$ is always a tangent to the parabola $y^2 = -4x$ for all non-zero values of $m$.
Statement $2$: Every tangent to the parabola $y^2 = -4x$ will meet its axis at a point whose abscissa is non-negative.

The equation of the parabola whose focus is $(6,0)$ and directrix is $x=-6$ is