The point on the parabola $y^2 = 8x$ at which the normal is parallel to the line $x - 2y + 5 = 0$ is

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 $P$?
$(A)$ $(4, 2\sqrt{2})$
$(B)$ $(9, 3\sqrt{2})$
$(C)$ $(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(D)$ $(1, \sqrt{2})$

The parametric equations of the parabola $y^2 - 12x - 2y - 11 = 0$ are:

Find the equation of the parabola that satisfies the following conditions: Focus $(6, 0)$,directrix $x = -6$.

What is the equation of the normal to the parabola $y^2 = 4ax$ at the point $(\frac{a}{m^2}, \frac{2a}{m})$?

The ends of the latus rectum of the conic ${x^2} + 10x - 16y + 25 = 0$ are