To find the equation of a parabola given its vertex and focus, we use the vertex form equation for a parabola. Since the focus and vertex are given, we'll determine whether the parabola opens upwards, downwards, left, or right, and then fit it into the standard form.
Given:
- Vertex [tex]\( V = (2, -1) \)[/tex]
- Focus [tex]\( F = (2, 3) \)[/tex]
### Step-by-Step Solution:
1. Determine the orientation of the parabola:
- The vertex and the focus have the same [tex]\( x \)[/tex]-coordinate ([tex]\(2\)[/tex]).
- The [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\(-1\)[/tex], and the [tex]\( y \)[/tex]-coordinate of the focus is [tex]\(3\)[/tex].
- Therefore, the vertex is below the focus, indicating that the parabola opens upwards.
2. Determine the distance [tex]\( p \)[/tex] (the focal length):
[tex]\( p \)[/tex] is the distance from the vertex to the focus along the axis of symmetry.
[tex]\[
p = \text{focus}_y - \text{vertex}_y = 3 - (-1) = 3 + 1 = 4
\][/tex]
3. Write the equation of the parabola:
The vertex form of a parabola that opens upwards is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Substituting [tex]\( h = 2 \)[/tex], [tex]\( k = -1 \)[/tex], and [tex]\( p = 4 \)[/tex]:
[tex]\[
(x - 2)^2 = 4 \cdot 4(y + 1)
\][/tex]
[tex]\[
(x - 2)^2 = 16(y + 1)
\][/tex]
4. Match the equation with the given options:
A. [tex]\( (x - 2)^2 = 16(y + 1) \)[/tex]
B. [tex]\( (x - 2)^2 = 4(y + 1) \)[/tex]
C. [tex]\( (x - 2)^2 = -16(y - 1) \)[/tex]
D. [tex]\( (x - 2) = 16(y + 1)^2 \)[/tex]
The equation derived, [tex]\( (x - 2)^2 = 16(y + 1) \)[/tex], matches with option A.
Therefore, the correct answer is
[tex]\[
\boxed{A}
\][/tex]
