Find the length of the subnormal of the parabola $y^2 = 16x$ at the point where the $x$-coordinate is $4$.

The locus of the mid-point of the line segment joining the focus and any point on the parabola $y^{2}=4ax$ is a parabola. Find the equation of its directrix.

Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(17, 289), (-2, 4), (13, 169)$,then the sum of the perpendicular distances from the directrix of $P$ to all four of the intersection points is:

If the line $y = 2x + k$ is a tangent to the curve $x^2 = 4y$,then $k$ is equal to

Find the measure of the angle in degrees subtended by the double ordinate of the parabola $y^2 = 4ax$ at its vertex.

If $P$ is $(3, 1)$ and $Q$ is a point on the curve $y^2 = 8x$,then the locus of the mid-point of the line segment $PQ$ is