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Find the equation for the following parabola:

Vertex [tex]\((2, -1)\)[/tex] Focus [tex]\((2, 3)\)[/tex]

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]


Sagot :

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]