To solve this problem, we need to identify key components of the given parabolic equation and then use those components to find the focus and the directrix.
The given equation of the parabola is:
[tex]\[
(y-4)^2 = 8(x-3)
\][/tex]
Step-by-Step Solution:
1. Identify the form of the parabolic equation:
The standard form for a horizontal parabola is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
Comparing this with the given equation [tex]\((y - 4)^2 = 8(x - 3)\)[/tex], we can match the components:
- [tex]\(h = 3\)[/tex]
- [tex]\(k = 4\)[/tex]
- [tex]\(4p = 8\)[/tex]
2. Solve for [tex]\(p\)[/tex]:
From the equation above, [tex]\(4p = 8\)[/tex]:
[tex]\[
p = \frac{8}{4} = 2
\][/tex]
3. Find the vertex:
The vertex [tex]\((h, k)\)[/tex] from the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is:
[tex]\[
(h, k) = (3, 4)
\][/tex]
4. Determine the focus:
For a horizontal parabola, the focus is located at [tex]\((h + p, k)\)[/tex]:
- [tex]\(h + p = 3 + 2 = 5\)[/tex]
- [tex]\(k = 4\)[/tex]
So, the coordinates of the focus are:
[tex]\[
(5, 4)
\][/tex]
5. Determine the directrix:
The directrix of a horizontal parabola is given by [tex]\(x = h - p\)[/tex]:
- [tex]\(h - p = 3 - 2 = 1\)[/tex]
Therefore, the equation of the directrix is:
[tex]\[
x = 1
\][/tex]
Thus, the solutions are:
- Focus: [tex]\((5, 4)\)[/tex]
- Directrix: [tex]\(x = 1\)[/tex]
