To find the focus and directrix of the given parabola, we'd look at its standard form and parameters. The given equation is:
[tex]\[
(y - 4)^2 = 16(x - 6)
\][/tex]
This equation is in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], with [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex] being parameters of the parabola.
1. Identify the vertex coordinates (h, k):
By comparing [tex]\((y - 4)^2 = 16(x - 6)\)[/tex] with [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we see that:
- [tex]\(k = 4\)[/tex]
- [tex]\(h = 6\)[/tex]
Therefore, the vertex of the parabola is at [tex]\((6, 4)\)[/tex].
2. Determine the value of [tex]\(p\)[/tex]:
From the standard form, [tex]\(4p\)[/tex] is the coefficient of [tex]\((x-h)\)[/tex]. Here, [tex]\(4p = 16\)[/tex]. Solving for [tex]\(p\)[/tex]:
[tex]\[
p = \frac{16}{4} = 4
\][/tex]
3. Find the coordinates of the focus:
The focus of a parabola given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is located at [tex]\((h + p, k)\)[/tex]. With [tex]\(h = 6\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(p = 4\)[/tex]:
[tex]\[
\text{Focus: } (h + p, k) = (6 + 4, 4) = (10, 4)
\][/tex]
4. Determine the directrix:
The directrix of a parabola given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is the line [tex]\(x = h - p\)[/tex]. With [tex]\(h = 6\)[/tex] and [tex]\(p = 4\)[/tex]:
[tex]\[
\text{Directrix: } x = h - p = 6 - 4 = 2
\][/tex]
Thus, the focus and the directrix of the parabola are:
[tex]\[
\text{Focus: } (10, 4)
\][/tex]
[tex]\[
\text{Directrix: } x = 2
\][/tex]
