To find the focus and directrix of the given parabola, let’s start by transforming and analyzing the given equation:
Given parabola:
[tex]\[
(x-5)^2 = 8(y-6)
\][/tex]
This equation is in the form [tex]\( (x-h)^2 = 4p(y-k) \)[/tex], which represents a parabola with a vertical axis. By comparing the given equation with the standard form, we identify the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(4p\)[/tex]:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 6\)[/tex]
- [tex]\(4p = 8\)[/tex]
To find [tex]\(p\)[/tex]:
[tex]\[
p = \frac{4p}{4} = \frac{8}{4} = 2.0
\][/tex]
Now we have identified the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 6\)[/tex]
- [tex]\(p = 2.0\)[/tex]
The focus of the parabola is found at [tex]\((h, k + p)\)[/tex]:
[tex]\[
\text{Focus} = (5, 6 + 2.0) = (5, 8.0)
\][/tex]
The directrix of the parabola is given by the line [tex]\( y = k - p \)[/tex]:
[tex]\[
\text{Directrix} = y = 6 - 2.0 = 4.0
\][/tex]
Therefore, the focus and directrix of the given parabola are:
[tex]\[
\text{Focus: } (5, 8.0)
\][/tex]
[tex]\[
\text{Directrix: } y = 4.0
\][/tex]
