At what points does the line $2x + y - 1 = 0$ intersect the parabola $y^2 = 4x$?

The coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is $4$ are:

Let $A_1, B_1, C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$,respectively. If $O=(0,0)$ and $C_1=(-4,0)$,then which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ The length of the line segment $OA_1$ is $4\sqrt{3}$
$(B)$ The length of the line segment $A_1 B_1$ is $16$
$(C)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(0,0)$
$(D)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(1,0)$

The point on the curve $x^{2}=2 y$ which is nearest to the point $(0,5)$ is

Two tangents are drawn from the point $(-1, -2)$ to the parabola $y^2 = 4x$. If $\theta$ is the angle between these tangents,then $\tan \theta = $

$PQ$ is any focal chord of the parabola $y^2=32x$. The length of $PQ$ can never be less than: ............ $unit$