The equation of a tangent to the parabola $y^2 = 8x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

Let $A$ be the vertex and $L$ be the length of the latus rectum of the parabola $y^2 - 2y - 4x - 7 = 0$. Find the equation of the parabola with $A$ as the vertex,$2L$ as the length of the latus rectum,and the axis at right angles to that of the given curve.

The lengths of the two focal chords of the parabola $y^2 = 16x$ are $25$ units each. If these two chords cut the parabola at $A, B, C$ and $D$,then the area (in sq. units) of the quadrilateral formed by $A, B, C$ and $D$ 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 $\Delta PQR$	$(S)$ $2$

The length of the chord of the parabola $x^2 = 4y$ having the equation $x - \sqrt{2}y + 4\sqrt{2} = 0$ is

If the line $lx + my + n = 0$ is tangent to the parabola $y^{2} = 4ax$,then