If the angle between the tangents drawn to the parabola $y^2 = 4x$ from a point on the line $4x - y = 0$ is $\frac{\pi}{3}$,then the sum of the abscissae of all such points is

If $A(at^2, 2at)$,$B(a/t^2, -2a/t)$,and $C(a, 0)$,then $2a$ is equal to

If $y=mx+4$ is a tangent to both the parabolas $y^{2}=4x$ and $x^{2}=2by$,then $b$ is equal to:

The parabola with directrix $x+2y-1=0$ and focus $(1,0)$ is

Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$,and let $Q$ and $Q^{\prime}$ be two distinct points on $E$ such that the lines $PQ$ and $PQ^{\prime}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) $TRUE$?
$(A)$ The triangle $PFQ$ is a right-angled triangle
$(B)$ The triangle $QPQ^{\prime}$ is a right-angled triangle
$(C)$ The distance between $P$ and $F$ is $5\sqrt{2}$
$(D)$ $F$ lies on the line joining $Q$ and $Q^{\prime}$

The number of normals drawn to the parabola $y^2=4x$ from the point $(1,0)$ is