To find the focus and directrix of the given parabola [tex]\((y + 3)^2 = 8(x + 3)\)[/tex], we will first rewrite this equation in the standard form of a parabola.
### Step 1: Identify the Standard Form
The standard form of a horizontal parabola that opens to the right is [tex]\((y - k)^2 = 4p(x - h)\)[/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 (also to the directrix).
### Step 2: Rewrite the Given Equation
The given equation is:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
We can rewrite this in the standard form by recognizing that:
[tex]\((y + 3)\)[/tex] is analogous to [tex]\((y - k)\)[/tex], and [tex]\((x + 3)\)[/tex] is analogous to [tex]\((x - h)\)[/tex].
Thus, comparing with [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
- [tex]\(k = -3\)[/tex]
- [tex]\(h = -3\)[/tex]
- [tex]\(4p = 8\)[/tex]
### Step 3: Solve for [tex]\(p\)[/tex]
From the equation [tex]\(4p = 8\)[/tex], we can solve for [tex]\(p\)[/tex]:
[tex]\[
4p = 8 \implies p = \frac{8}{4} = 2
\][/tex]
### Step 4: Find the Focus
The focus of a parabola in this form is located at [tex]\((h + p, k)\)[/tex].
Substituting the known values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
[tex]\[
(h + p, k) = (-3 + 2, -3) = (-1, -3)
\][/tex]
Thus, the coordinates of the focus are [tex]\((-1, -3)\)[/tex].
### Step 5: Find the Directrix
The directrix of the parabola in this form is given by the vertical line:
[tex]\[
x = h - p
\][/tex]
Substituting the known values of [tex]\(h\)[/tex] and [tex]\(p\)[/tex]:
[tex]\[
x = -3 - 2 = -5
\][/tex]
Thus, the equation of the directrix is [tex]\(x = -5\)[/tex].
### Final Answer
- Focus: [tex]\((-1, -3)\)[/tex]
- Directrix: [tex]\(x = -5\)[/tex]
So, the answer to the question is:
Focus: ([–1], [tex]\(-3\)[/tex])
Directrix: [tex]\(x = -5\)[/tex]
