To determine which equation represents the parabola with a vertex at [tex]\((4, 1)\)[/tex] and directrix [tex]\(y = 6\)[/tex], we can follow these steps:
1. Understand the standard form of a parabola:
The standard equation of a parabola that opens vertically (either up or down) with a vertex at [tex]\((h, k)\)[/tex] is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
where [tex]\(p\)[/tex] is the distance from the vertex to the focus (or directrix, depending on the direction of the parabola).
2. Determine the value of [tex]\(p\)[/tex]:
Since the directrix [tex]\(y = 6\)[/tex] is vertical and above the vertex [tex]\((4, 1)\)[/tex], we calculate [tex]\(p\)[/tex], which is the distance between the vertex and the directrix:
[tex]\[
p = k - \text{directrix\_y} = 1 - 6 = -5
\][/tex]
The negative sign indicates that the parabola opens downward.
3. Substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex] into the standard form:
Given [tex]\(h = 4\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = -5\)[/tex], we substitute these values into the standard form equation:
[tex]\[
(x - 4)^2 = 4(-5)(y - 1)
\][/tex]
Simplify the equation:
[tex]\[
(x - 4)^2 = -20(y - 1)
\][/tex]
4. Match the derived equation with the given choices:
The equation we derived is [tex]\((x - 4)^2 = -20(y - 1)\)[/tex]. Comparing with the given choices, we see that the correct match is:
[tex]\[
(x - 4)^2 = -20(y - 1)
\][/tex]
Therefore, the equation that represents the parabola with vertex [tex]\((4, 1)\)[/tex] and directrix [tex]\(y = 6\)[/tex] is:
[tex]\[
\boxed{(x - 4)^2 = -20(y - 1)}
\][/tex]
