To find the focus and directrix of the given parabola equation:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
we need to transform it into its standard form and identify the key components. Here's a step-by-step solution:
### Step 1: Identify the Standard Form
The given equation of the parabola is:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
This equation resembles the standard form of a horizontally oriented parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola and [tex]\(p\)[/tex] determines the distance from the vertex to the focus and the directrix.
### Step 2: Identify the Vertex (h, k)
The equation [tex]\((y + 3)^2 = 8(x + 3)\)[/tex] can be rewritten in vertex form:
[tex]\[
(y - (-3))^2 = 4p(x - (-3))
\][/tex]
From this, we can see that:
- The vertex [tex]\((h, k)\)[/tex] is [tex]\((-3, -3)\)[/tex].
### Step 3: Determine the Value of p
From the standard form equation [tex]\((y - k)^2 = 4p(x - h)\)[/tex],
[tex]\[4p = 8\][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[4p = 8 \implies p = \frac{8}{4} = 2\][/tex]
### Step 4: Find the Coordinates of the Focus
The focus of a horizontally oriented parabola is given by [tex]\((h + p, k)\)[/tex].
Given:
[tex]\[h = -3, \quad k = -3, \quad p = 2\][/tex]
Thus, the focus is:
[tex]\[
(h + p, k) = (-3 + 2, -3) = (-1, -3)
\][/tex]
### Step 5: Find the Equation of the Directrix
The directrix of a horizontally oriented parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is given by [tex]\(x = h - p\)[/tex].
Given:
[tex]\[h = -3, \quad p = 2\][/tex]
Thus, the directrix is:
[tex]\[
x = h - p = -3 - 2 = -5
\][/tex]
### Final Answer:
- Focus: [tex]\((-1, -3)\)[/tex]
- Directrix: [tex]\(x = -5\)[/tex]
