The axis of a parabola lies along the $x$-axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin on the positive $x$-axis,then which of the following points does not lie on it?

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the endpoints of its latus rectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola,and $\Delta_2$ be the area of the triangle formed by the intersection points of the tangents drawn at $P$ and at the endpoints of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is

If two parabolas $y^2 = 4x$ and $x^2 = 32y$ intersect at the point $(16, 8)$ at an angle $\theta$,then what is the value of $\tan \theta$?

How many parabolas can be drawn if the two endpoints of the latus rectum are given?

The point of contact of the tangent $18x - 6y + 1 = 0$ to the parabola $y^2 = 2x$ is

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $x$-axis is