The equation of the parabola with focus $(0,0)$ and directrix $x+y=4$ is

If the line $y=2x+k$ is a normal to the parabola $y^2=4x$,then $k=$

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is $3\sqrt{2}$,then which of the following is (are) the coordinates of $P$?
$(A)$ $(4, 2\sqrt{2})$
$(B)$ $(9, 3\sqrt{2})$
$(C)$ $(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(D)$ $(1, \sqrt{2})$

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:

If one end of a focal chord of the parabola $y^2 = \frac{8}{a} x$ $(a > 0)$ is at $(1, 4)$,then the length of this focal chord is

The distance of the focus of the parabola $y^2 = 16x$ from its directrix is...