The line $y = 2x + c$ is tangent to the parabola $y^2 = 4x$,then $c = $

Find the equation of the normal to the curve $x^{2}=4y$ which passes through the point $(1,2)$.

The line $y - \sqrt{3}x + 3 = 0$ cuts the parabola $y^2 = x + 2$ at the points $P$ and $Q$. If the coordinates of the point $X$ are $(\sqrt{3}, 0)$, then the value of $XP \cdot XQ$ is

The locus of the midpoint of the chord of the parabola $y^2 = 4ax$ which subtends a right angle at the vertex is

If $x-2y+k=0$ is a tangent to the parabola $y^2-4x-4y+8=0$,then the slope of the tangent drawn at $(1, k)$ on the given parabola is

Suppose $AB$ is a focal chord of the parabola $y^2=12x$ of length $l$ and slope $m < \sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$,then $l \cdot d^2$ is equal to ....................