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

If the parabolas $y^2 = 4b(x - c)$ and $y^2 = 8ax$ have a common normal,then which one of the following is a valid choice for the ordered triad $(a, b, c)$?

The vertex of the parabola $(y - 2)^2 = 16(x - 1)$ is

Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2,0)$ to the parabola $2y^{2} = -x$. If the lines $l_{1}$ and $l_{2}$ are also tangent to the circle $(x-5)^{2} + y^{2} = r$,then $17r$ is equal to.

Find the length of the latus rectum of the parabola $y = x \tan \alpha - \frac{g x^2}{2 u^2 \cos^2 \alpha}$.

If the vertex of a parabola is $(4,3)$ and its directrix is $3x+2y-7=0$,then the equation of the latus rectum of the parabola is