To find the focus and directrix of the given parabola [tex]\( (y - 6)^2 = 12(x - 2) \)[/tex], let's start by understanding its standard form and some properties of the parabola.
Given equation:
[tex]\[ (y - 6)^2 = 12(x - 2) \][/tex]
This is of the form [tex]\( (y - k)^2 = 4p(x - h) \)[/tex], where:
- [tex]\((h, k)\)[/tex] is the vertex of the parabola.
- [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also the distance from the vertex to the directrix).
### Step-by-Step Solution:
1. Identify the vertex:
From the given equation [tex]\( (y - 6)^2 = 12(x - 2) \)[/tex], we can see that:
[tex]\[
h = 2, \quad k = 6
\][/tex]
Thus, the vertex of the parabola is [tex]\( (2, 6) \)[/tex].
2. Find the value of [tex]\( p \)[/tex]:
The given equation [tex]\( (y - 6)^2 = 12(x - 2) \)[/tex] can be compared to the standard form [tex]\( (y - k)^2 = 4p(x - h) \)[/tex]. Here, we have:
[tex]\[
4p = 12
\][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{12}{4} = 3
\][/tex]
3. Determine the focus:
For a parabola of the form [tex]\( (y - k)^2 = 4p(x - h) \)[/tex], the focus lies at [tex]\( (h + p, k) \)[/tex].
Given [tex]\( h = 2 \)[/tex], [tex]\( k = 6 \)[/tex], and [tex]\( p = 3 \)[/tex]:
[tex]\[
\text{Focus} = (h + p, k) = (2 + 3, 6) = (5, 6)
\][/tex]
4. Determine the directrix:
The directrix of a parabola [tex]\( (y - k)^2 = 4p(x - h) \)[/tex] is the vertical line given by [tex]\( x = h - p \)[/tex].
Given [tex]\( h = 2 \)[/tex] and [tex]\( p = 3 \)[/tex]:
[tex]\[
\text{Directrix} = x = h - p = 2 - 3 = -1
\][/tex]
### Summary:
Focus: [tex]\( (5, 6) \)[/tex]
Directrix: [tex]\( x = -1 \)[/tex]
