The latus rectum is the line segment through a focus of a conic section, perpendicular to the major axis, which has both endpoints on the curve.
The length of a parabola's latus rectum is $4p$, where $p$ is the distance from the focus to the vertex.
Since the length of the latus rectum of your parabola is the length of the line segment joining the points $(2, 4)$ and $(6, 4)$, i.e. four units, then from $4p = 4$, so $p$ must be $1$. The focus will be the midpoint of the latus rectum: focus = $(4, 4)$.
Your parabola is "vertical", since the latus rectum lies on the line $y = 4$ (parallel to the x-axis). Since the point $(8, 1)$ also lies on your parabola, you should be able to tell that the parabola must open downward (in the negative $y$ direction).
With this information, and knowing $p =1 $ with $p$ being the distance from the focus $(4, 4)$, you should be able to determine that the vertex $(h, k) = (5, 4)$ of your parabola (since it's one unit "up" (in positive y direction) from the focus $(4, 4))$.
Use this information to determine the equation for the parabola using the formula for a vertical parabola opening downward: $$-4p(y-k) = (x - h)^2.$$