To find the equation of a parabola given its vertex and directrix, we need to follow these steps:
1. Identify the standard form of the parabola:
The standard form of a parabola with a horizontal orientation and vertex [tex]\((h, k)\)[/tex] is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
where [tex]\(p\)[/tex] is the distance from the vertex to the focus (or equivalently, from the vertex to the directrix, but with opposite sign).
2. Substitute the vertex coordinates:
Given the vertex is [tex]\((0, 5)\)[/tex], we have [tex]\(h = 0\)[/tex] and [tex]\(k = 5\)[/tex]. So, the equation becomes:
[tex]\[
(y - 5)^2 = 4p(x - 0)
\][/tex]
which simplifies to:
[tex]\[
(y - 5)^2 = 4px
\][/tex]
3. Determine the value of [tex]\(p\)[/tex]:
The distance from the vertex to the directrix [tex]\(x = 2\)[/tex] is calculated as the absolute difference in the x-coordinates:
[tex]\[
|0 - 2| = 2
\][/tex]
Since the directrix is to the right of the vertex, the parabola opens to the left, and therefore [tex]\(p\)[/tex] is negative. Thus, [tex]\(p = -2\)[/tex].
4. Substitute [tex]\(p\)[/tex] into the equation:
Now we substitute [tex]\(p = -2\)[/tex] into the equation:
[tex]\[
(y - 5)^2 = 4(-2)x
\][/tex]
which simplifies to:
[tex]\[
(y - 5)^2 = -8x
\][/tex]
Therefore, the equation of the parabola that fits the given conditions is:
[tex]\[
(y - 5)^2 = -8x
\][/tex]
Thus, the correct answer is:
[tex]\[ \text{D. } (y - 5)^2 = -8x \][/tex]
