To find the equation of a parabola with a given focus and directrix, we follow these steps:
1. Identify the focus and directrix:
- Focus: [tex]\((-2, 4)\)[/tex]
- Directrix: [tex]\(y = 6\)[/tex]
2. Determine the position of the vertex:
The vertex of the parabola lies midway between the focus and the directrix. To find the [tex]\(y\)[/tex]-coordinate of the vertex, we average the [tex]\(y\)[/tex]-coordinate of the focus and the value of the directrix:
[tex]\[
\text{vertex}_y = \frac{4 + 6}{2} = 5
\][/tex]
The [tex]\(x\)[/tex]-coordinate of the vertex is the same as the [tex]\(x\)[/tex]-coordinate of the focus, which is [tex]\(-2\)[/tex]. Thus, the coordinates of the vertex are:
[tex]\[
\text{vertex} = (-2, 5)
\][/tex]
3. Determine the value of [tex]\( p \)[/tex]:
The value of [tex]\( p \)[/tex] is the distance from the vertex to the focus (or from the vertex to the directrix). This distance is:
[tex]\[
p = \text{vertex}_y - \text{focus}_y = 5 - 6 = -1
\][/tex]
Since the directrix is above the vertex, [tex]\( p \)[/tex] is negative, indicating that the parabola opens downwards.
4. Write the equation of the parabola in vertex form:
The standard form of a parabola with vertex [tex]\((h, k)\)[/tex] and value [tex]\( p \)[/tex] is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
Substituting [tex]\(h = -2\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(p = -1\)[/tex], we get:
[tex]\[
(x - (-2))^2 = 4(-1)(y - 5)
\][/tex]
Simplifying, we get the final equation:
[tex]\[
(x + 2)^2 = -4(y - 5)
\][/tex]
Hence, the equation of the parabola is:
[tex]\[
(x + 2)^2 = -4(y - 5)
\][/tex]
