Answer: (x-4)^2 = 8(y-1)
Step-by-step explanation: To write the equation of a parabola given the directrix and vertex, we need to determine the equation's specific form.
Since the directrix is x=2 and the vertex is (4,1), we can conclude that the parabola opens horizontally to the left or right. This is because the directrix is a vertical line, meaning the parabola's axis of symmetry is vertical.
To find the equation of the parabola, we can use the standard form of the equation for a parabola with a horizontal axis of symmetry:
(x-h)^2 = 4p(y-k)
Where (h,k) is the vertex and p is the distance between the vertex and the focus or the directrix.
In this case, the vertex is (4,1), so we have (h,k) = (4,1). The directrix is x=2, so p is the distance between the vertex and x=2, which is 2 units.
Plugging these values into the equation, we have:
(x-4)^2 = 4(2)(y-1)
Simplifying further:
(x-4)^2 = 8(y-1)
This is the equation of the parabola with a directrix of x=2 and a vertex of (4,1).