Find the angle of intersection between the curves $y^2 = 4ax$ and $x^2 = 4ay$.

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,passing through $(5, 2)$,and symmetric with respect to the $y$-axis.

The largest value of $k$ for which the circle $x^2+y^2=k^2$ lies completely in the interior of the parabola $y^2=4x+16$ 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 $P$?
$(A)$ $(4, 2\sqrt{2})$
$(B)$ $(9, 3\sqrt{2})$
$(C)$ $(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(D)$ $(1, \sqrt{2})$

Find the length of the tangent drawn from an endpoint of the latus rectum of the parabola $y^2 = 4ax$ to a circle of radius $a$ that touches the parabola at its vertex.

The normal at the point $(2, 4)$ to the parabola $y^2 = 4ax$ meets the parabola again at another point. Find the coordinates of this point.