If the parabolas $y^2 = 4b(x - c)$ and $y^2 = 8ax$ have a common normal,then which one of the following is a valid choice for the ordered triad $(a, b, c)$?

The vertex of the parabola is at $(1, 2)$ and its axis is parallel to the $y$-axis. If the parabola passes through $(0, 6)$,then its latus rectum is:

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$,respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
$(A)$ vertex is $\left(\frac{2a}{3}, 0\right)$
$(B)$ directrix is $x=0$
$(C)$ latus rectum is $\frac{2a}{3}$
$(D)$ focus is $(a, 0)$

The point on the axis of the parabola $3y^2+4y-6x+8=0$ from which $3$ real normals can be drawn is given by:

Two perpendicular tangents to the parabola $y^2 = 4ax$ always intersect on the line:

Find the shortest distance of the point $(0, c)$ from the parabola $y=x^{2}$ where $0 \leq c \leq 5$.