To find the equation of the parabola with its focus at [tex]$(-5, 0)$[/tex] and its directrix [tex]$y=2$[/tex], let's go through the following steps:
1. Identify the Vertex of the Parabola:
The vertex of the parabola is halfway between the focus and the directrix. Since the focus's y-coordinate is 0 and the directrix is the line [tex]$y=2$[/tex], the y-coordinate of the vertex is the average of these two y-coordinates.
[tex]\[
\text{vertex}_y = \frac{0 + 2}{2} = 1
\][/tex]
The x-coordinate of the vertex is the same as that of the focus since the directrix is horizontal.
[tex]\[
\text{vertex}_x = -5
\][/tex]
Therefore, the vertex of the parabola is at [tex]$(-5, 1)$[/tex].
2. Determine the Value of 'a':
The formula for the value of 'a' in the standard form of a parabola [tex]\(y = a(x - h)^2 + k\)[/tex] is given by [tex]$a = \frac{1}{4p}$[/tex], where [tex]\(p\)[/tex] is the distance from the vertex to the focus or the directrix.
Calculate [tex]\(p\)[/tex]:
[tex]\[
p = \text{distance from vertex to focus} = |1 - 0| = 1
\][/tex]
Thus,
[tex]\[
a = \frac{1}{4p} = \frac{1}{4 \times 1} = \frac{1}{4}
\][/tex]
3. Write the Equation of the Parabola:
Using the vertex [tex]\((-5, 1)\)[/tex] and the value of [tex]\(a = \frac{1}{4}\)[/tex], the equation of the parabola is:
[tex]\[
y = a(x - (-5))^2 + 1
\][/tex]
Simplify it:
[tex]\[
y = \frac{1}{4}(x + 5)^2 + 1
\][/tex]
Therefore, from the given options, the correct equation for the parabola is:
[tex]\[
\boxed{D) \ y= \frac{1}{4}(x+5)^2+1}
\][/tex]
