If $y = 2x + 3$ is a tangent to the parabola $y^2 = 24x$,what is the distance between the tangent and the parallel normal?

The area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12y$ to the extremities of its latus rectum is

The area of the triangle formed by the lines joining the vertex of the parabola $x^2=12y$ to the ends of its latus rectum is equal to $...$ sq. units.

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 $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$,then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

The parabola with directrix $x+2y-1=0$ and focus $(1,0)$ is