If the focus of a parabola is at the origin $(0, 0)$ and the line $x = 2$ is its directrix,what is the vertex of the parabola?

The locus of the foot of the perpendicular drawn from the focus to any tangent of the parabola $y^2 = 4ax$ is:

Let $P(at^{2}, 2at)$, $Q$, and $R(ar^{2}, 2ar)$ be three points on the parabola $y^{2}=4ax$. If $PQ$ is a focal chord and $PK$ is parallel to $QR$, where the coordinates of $K$ are $(2a, 0)$, then the value of $r$ is:

What is the distance between the focus and the directrix of the parabola $x^2 = -8y$?

If the tangents and normals at the extremities of a focal chord of a parabola $y^2 = 4ax$ intersect at $(x_1, y_1)$ and $(x_2, y_2)$ respectively,then:

$A$ line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan^{-1}\left(\frac{3}{4}\right)$ with the $X$-axis. It intersects the parabola $y^{2}=4(x-3)$ at points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, respectively. Then $|x_{1}-x_{2}|$ is equal to