The diameter of the parabola $y^2 = 4ax$ which bisects the chords parallel to $y = mx + \alpha$ is:

The equation of the locus of all points equidistant from the point $(4, 2)$ and the $x$-axis is:

Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2,0)$ to the parabola $2y^{2} = -x$. If the lines $l_{1}$ and $l_{2}$ are also tangent to the circle $(x-5)^{2} + y^{2} = r$,then $17r$ is equal to.

If the normals at two points $P$ and $Q$ of a parabola $y^2 = 4ax$ intersect at a third point $R$ on the curve,then the product of the ordinates of $P$ and $Q$ is (in $a^2$)

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$,such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant,enclosed by the parabola $y^2=x$,the curve $S$,and the lines $x=1$ and $x=4$. Then which of the following statements is (are) True?
$(A) \ (4, \sqrt{3}) \in S$
$(B) \ (5, \sqrt{2}) \in S$
$(C)$ Area of $R$ is $\frac{14}{3}-2 \sqrt{3}$
$(D)$ Area of $R$ is $\frac{14}{3}-\sqrt{3}$

The shortest distance from $(0,3)$ to the parabola $y^2=4x$ is