The normal meets the parabola $y^2 = 4ax$ at a point where the abscissa is equal to the ordinate. Find this point.

The number of normals drawn to the parabola $y^2=4x$ from the point $(1,0)$ is

If a normal is drawn to the parabola $y^2 = 4ax$ at the point $(2a, 2\sqrt{2}a)$,what is the length of the normal chord?

The equation of the chord of contact of the tangents drawn from the point $(2, 3)$ to the parabola $y^2 + x = 0$ is:

The equation of the directrix of the parabola $x^2+8x+12y+4=0$ 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 $P$?
$(A)$ $(4, 2\sqrt{2})$
$(B)$ $(9, 3\sqrt{2})$
$(C)$ $(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(D)$ $(1, \sqrt{2})$