The tangents to the parabola $y^2 = 4ax$ from an external point $P$ make angles $\theta_1$ and $\theta_2$ with the axis of the parabola,such that $\tan \theta_1 + \tan \theta_2 = b$,where $b$ is a constant. Then $P$ lies on

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

Find the equations of the tangent and normal to the parabola $y^{2}=4ax$ at the point $(at^{2}, 2at)$.

The equation of the parabola whose focus is $(5, 3)$ and directrix is $3x - 4y + 1 = 0$ is:

What is the maximum number of normals that can be drawn from any interior point to a parabola?

If $(2,3)$ is the focus and $x-y+3=0$ is the directrix of a parabola,then the equation of the tangent drawn at the vertex of the parabola is