To find the equation of a parabola with the focus at [tex]\((4,6)\)[/tex] and the directrix at [tex]\(y = 2\)[/tex], we can follow these steps:
1. Find the vertex of the parabola:
- The vertex lies exactly midway between the focus and the directrix.
- The y-coordinate of the vertex, [tex]\(k\)[/tex], is the average of the y-coordinate of the focus and the y-coordinate of the directrix:
[tex]\[
k = \frac{focus\_y + directrix\_y}{2} = \frac{6 + 2}{2} = 4.
\][/tex]
- The x-coordinate of the vertex, [tex]\(h\)[/tex], is the same as the x-coordinate of the focus:
[tex]\[
h = 4.
\][/tex]
- Therefore, the vertex of the parabola is [tex]\((4,4)\)[/tex].
2. Determine the value of [tex]\(p\)[/tex], where [tex]\(p\)[/tex] is the distance from the vertex to the focus (or directrix):
- To find [tex]\(p\)[/tex], subtract the y-coordinate of the vertex from the y-coordinate of the focus:
[tex]\[
p = focus\_y - vertex\_y = 6 - 4 = 2.
\][/tex]
- Hence, [tex]\(p = 2\)[/tex].
3. Write the equation of the parabola:
- A parabola with a vertical axis of symmetry is given by the equation [tex]\((x - h)^2 = 4p(y - k)\)[/tex].
- Here, [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the distance we calculated.
- Substituting [tex]\(h = 4\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(p = 2\)[/tex]:
[tex]\((x - 4)^2 = 4 \cdot 2 (y - 4)\)[/tex]
Simplifying,
[tex]\[
(x - 4)^2 = 8(y - 4)
\][/tex]
So, the equation of the parabola is:
[tex]\[
(x - 4)^2 = 8(y - 4)
\][/tex]
