To determine the equation of the parabola with a focus at [tex]\((-6, 0)\)[/tex] and a directrix of [tex]\(y = 8\)[/tex], follow these steps:
1. Identify the Vertex:
The vertex is halfway between the focus and the directrix. The focus has coordinates [tex]\((-6, 0)\)[/tex], and the directrix is a horizontal line given by [tex]\(y = 8\)[/tex]. The vertex y-coordinate is calculated as the average of the focus y-coordinate and the directrix y-coordinate:
[tex]\[
\text{Vertex y-coordinate} = \frac{0 + 8}{2} = 4
\][/tex]
Since the focus is at [tex]\((-6, 0)\)[/tex] and the directrix does not change the x-coordinate, the vertex's x-coordinate remains [tex]\(-6\)[/tex]. Therefore, the vertex is at [tex]\((-6, 4)\)[/tex].
2. Determine the Value of [tex]\(p\)[/tex]:
The value [tex]\(p\)[/tex] represents the distance from the vertex to the focus. In this case:
[tex]\[
p = 4 - 0 = 4
\][/tex]
Since the parabola opens downwards (the focus is below the directrix), the equation of the parabola in vertex form is given by:
[tex]\[
y = -\frac{1}{4p}(x - h)^2 + k
\][/tex]
3. Substitute the Values into the Equation:
Given [tex]\(h = -6\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(p = 4\)[/tex]:
[tex]\[
y = -\frac{1}{4 \cdot 4}(x + 6)^2 + 4
\][/tex]
Simplify the equation:
[tex]\[
y = -\frac{1}{16}(x + 6)^2 + 4
\][/tex]
Thus, the equation of the parabola is:
[tex]\[
y = -\frac{1}{16}(x + 6)^2 + 4
\][/tex]
Comparing this with the given options, the correct answer is:
[tex]\[
\boxed{y = -\frac{1}{16}(x+6)^2 + 4}
\][/tex]
