To find the focus and directrix of the given parabola [tex]\((x-3)^2 = 4(y-3)\)[/tex], let's follow a detailed, step-by-step approach.
### Step 1: Identify the Standard Form
The given equation [tex]\((x-3)^2 = 4(y-3)\)[/tex] fits the standard form of a vertical parabola:
[tex]\[
(x-h)^2 = 4p(y-k)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(p\)[/tex] is the distance from the vertex to the focus (or directrix).
### Step 2: Determine the Vertex
From the equation [tex]\((x-3)^2 = 4(y-3)\)[/tex], we can see that [tex]\(h = 3\)[/tex] and [tex]\(k = 3\)[/tex]. Therefore, the vertex of the parabola is [tex]\((3, 3)\)[/tex].
### Step 3: Identify the Value of [tex]\(p\)[/tex]
The coefficient [tex]\(4p\)[/tex] in the equation [tex]\((x-3)^2 = 4(y-3)\)[/tex] is equal to 4. Thus, we can determine [tex]\(p\)[/tex] by solving the equation:
[tex]\[
4p = 4 \implies p = 1
\][/tex]
### Step 4: Locate the Focus
The focus of a vertical parabola [tex]\((x-h)^2 = 4p(y-k)\)[/tex] lies [tex]\(p\)[/tex] units above the vertex (since [tex]\(p\)[/tex] is positive). Therefore, the coordinates of the focus are:
[tex]\[
\text{Focus} = (h, k + p) = (3, 3 + 1) = (3, 4)
\][/tex]
### Step 5: Determine the Directrix
The directrix of a vertical parabola [tex]\((x-h)^2 = 4p(y-k)\)[/tex] is a horizontal line [tex]\(p\)[/tex] units below the vertex. Thus, the equation of the directrix is:
[tex]\[
\text{Directrix} = y = k - p = 3 - 1 = 2
\][/tex]
### Summary
Putting it all together, we have:
- Focus: [tex]\((3, 4)\)[/tex]
- Directrix: [tex]\(y = 2\)[/tex]
So, the final answers are:
Focus: [tex]\(( [3], \square 4 )\)[/tex]
Directrix: [tex]\(y = \square 2\)[/tex]
