To find the equation of the parabola with vertex [tex]\((3,4)\)[/tex] and directrix [tex]\(y=2\)[/tex], let's follow these steps:
1. Determine the Focus:
- The vertex of the parabola is given as [tex]\((3, 4)\)[/tex].
- The directrix is [tex]\(y = 2\)[/tex].
- The focus and the directrix are equidistant from the vertex. Knowing that the vertex y-value is greater than the y-value of the directrix by [tex]\(4 - 2 = 2\)[/tex] units, the focus must be 2 units above the vertex.
- Thus, the focus is at [tex]\((3, 4 + 2) = (3, 6)\)[/tex].
2. Determine the Distance [tex]\(p\)[/tex]:
- [tex]\(p\)[/tex] is the distance from the vertex to the focus (or equivalently to the directrix, but in the opposite direction):
[tex]\[
p = 4 - 2 = 2
\][/tex]
3. Use the Standard Parabola Equation:
- The standard form for a parabola with a vertical axis of symmetry (i.e., opening up or down) is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
4. Plug in the Values:
- The vertex [tex]\((h, k) = (3, 4)\)[/tex].
- [tex]\(p = 2\)[/tex].
Substituting these into the standard form equation:
[tex]\[
(x - 3)^2 = 4 \cdot 2 \cdot (y - 4)
\][/tex]
Simplify the equation:
[tex]\[
(x - 3)^2 = 8(y - 4)
\][/tex]
So, the correct equation of the parabola is:
[tex]\[
(x - 3)^2 = 8(y - 4)
\][/tex]
Based on the options given, the correct choice is:
[tex]\[
\boxed{(x - 3)^2 = 8(y - 4)}
\][/tex]
Thus, the correct answer is Option A: [tex]\((x - 3)^2 = 8(y - 4)\)[/tex].
