What is the equation of the normal to the curve $y^2 = 16x$ at the point $(1, 4)$?

Find the locus of the point of contact of the tangent to the parabola $y^2 = 4x$,where the tangent makes an angle of $45^{\circ}$ with the $x$-axis.

The length of the latus rectum of the conic $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ is

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 $\Delta PQR$	$(S)$ $2$

What is the condition for the points $(a, 0)$,$(at_1^2, 2at_1)$,and $(at_2^2, 2at_2)$ to be collinear?

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: