To determine the equation of the parabola with a given focus and directrix, let's follow a step-by-step approach.
### Step 1: Identify Given Information
- Focus: [tex]\((0, -2)\)[/tex]
- Directrix: [tex]\(y = 2\)[/tex]
### Step 2: Determine the Parabola's Characteristics
For a parabola:
- The directrix is equidistant from the vertex as the focus is from the vertex.
- The general vertex form of the parabola is [tex]\((x - h)^2 = 4p(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also from the vertex to the directrix).
### Step 3: Setup Equations Based on Given Focus and Directrix
Given the focus [tex]\((0, -2)\)[/tex] and directrix [tex]\(y = 2\)[/tex], we can extract the following:
- Vertex: The vertex [tex]\((h, k)\)[/tex] lies exactly halfway between the directrix and the focus along the vertical line (since the x-coordinate of the focus matches the symmetry line).
Thus, the vertex [tex]\(k\)[/tex] is the midpoint between [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[
k = \frac{-2 + 2}{2} = 0
\][/tex]
- Therefore, vertex [tex]\((0, 0)\)[/tex]
### Step 4: Calculate [tex]\(p\)[/tex]
- Distance [tex]\(p\)[/tex] from the vertex to the focus (or directrix):
[tex]\( |p| = \text{distance from } (0,0) \text{ to } (0,-2) = 2 \)[/tex]
Since the focus is below the vertex, [tex]\(p = -2\)[/tex].
### Step 5: Formulate the Equation
Substitute [tex]\(h = 0\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(p = -2\)[/tex] into the general vertex form [tex]\((x - h)^2 = 4p(y - k)\)[/tex]:
[tex]\[
(x - 0)^2 = 4(-2)(y - 0)
\][/tex]
[tex]\[
x^2 = -8y
\][/tex]
### Step 6: Compare with Given Options
The resulting equation, [tex]\(x^2 = -8y\)[/tex], matches the form of one of the provided options.
### Conclusion
The equation that represents the parabola is:
\[
x^2 = -8y
\
