The length of the latus rectum of a parabola whose focal chord $PSQ$ is such that $PS = 3$ and $QS = 2$ is

If $PSQ$ is a focal chord of the parabola $y^2 = 8x$ such that $SP = 6$,then find the length of $SQ$.

If the segment intercepted by the parabola $y^2 = 4ax$ with the line $lx + my + n = 0$ subtends a right angle at the vertex,then

The equation of the common tangent of the parabolas $x^2=108y$ and $y^2=32x$ is

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 $(1,0)$

Statement $1$: $y = mx - \frac{1}{m}$ is always a tangent to the parabola $y^2 = -4x$ for all non-zero values of $m$.
Statement $2$: Every tangent to the parabola $y^2 = -4x$ will meet its axis at a point whose abscissa is non-negative.