If the focus divides a focal chord of the parabola $y^2 = 16x$ into $2$ parts having lengths $a$ and $c$,such that $a, b, c$ are in $H.P.$,then the value of $b$ is equal to:

If the focus of a parabola is $(3, -4)$ and its directrix is $x + y - 2 = 0$,find the vertex.

Let $P$ be the point $(2, 0)$ and $Q$ be a variable point on the parabola $(y - 6)^2 = 2(x - 4)$. Then the locus of the mid-point of $PQ$ is:

$S \equiv y^2 - 4ax = 0$ and $S' \equiv y^2 + ax = 0$ are two parabolas,and $P(t)$ is a point on the parabola $S' = 0$. If $A$ and $B$ are the feet of the perpendiculars from $P$ onto the coordinate axes and $AB$ is a tangent to the parabola $S = 0$ at the point $Q(t_1)$,then $t_1 =$

Three normals are drawn from the point $(3, 0)$ to the parabola $y^2 = 4x$,meeting the parabola at points $P, Q,$ and $R$. Match the following:

Column-$I$	Column-$II$
$(A)$ Circumradius of $\Delta PQR$	$(P)$ $5/2$
$(B)$ Area of $\Delta PQR$	$(Q)$ $(5/2, 0)$
$(C)$ Centroid of $\Delta PQR$	$(R)$ $(2/3, 0)$
$(D)$ Circumcenter of $\Delta PQR$	$(S)$ $2$

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin,respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$,then the area (in $sq. \text{ units}$) of $\triangle SOR$ is equal to: