Tangents drawn at the ends of any focal chord of a parabola $y^2 = 4ax$ intersect on the line

If the focus of a parabola is $(3, -4)$ and its directrix is $x + y - 2 = 0$,find the vertex.

Let $P(x, y)$ be the midpoint of the line joining $(1, 0)$ to a point on the curve $y^{2} = \left|\begin{array}{ll}x+1 & x+2 \\ x+3 & x+5\end{array}\right|$. Then,the locus of $P$ is symmetrical about

The locus of the poles of focal chords of a parabola is:

Let $O$ be the vertex and $Q$ be any point on the parabola $x^2=8y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then the locus of $P$ is

The tangents drawn from the ends of the latus rectum of the parabola ${y^2} = 12x$ meet at: