To find the focus and directrix of the parabola given by the equation:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
we will follow these steps:
### Step 1: Rewrite the Given Parabola Equation
First, let's rewrite the given equation in a more familiar form. The given equation is:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
### Step 2: Identify the Standard Form and Parameters
We recognize that this equation resembles the standard form of a parabola that opens horizontally:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
### Step 3: Compare with Standard Form
By comparing:
[tex]\[
(y + 3)^2 = 8(x + 3)
\][/tex]
to:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
we can see that:
- [tex]\( k = -3 \)[/tex]
- [tex]\( h = -3 \)[/tex]
- [tex]\( 4p = 8 \)[/tex]
### Step 4: Solve for 'p'
Now, solve for [tex]\( p \)[/tex]:
[tex]\[
4p = 8 \implies p = \frac{8}{4} = 2
\][/tex]
### Step 5: Determine the Coordinates of the Focus
The focus of the parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is given by the coordinates [tex]\((h + p, k)\)[/tex].
With the values substituted in:
[tex]\[
h = -3, \quad k = -3, \quad p = 2
\][/tex]
So the coordinates of the focus are:
[tex]\[
(h + p, k) = (-3 + 2, -3) = (-1, -3)
\][/tex]
### Step 6: Find the Equation of the Directrix
The directrix of the parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex] has the equation:
[tex]\[
x = h - p
\][/tex]
With the values substituted in:
[tex]\[
h = -3, \quad p = 2
\][/tex]
The equation of the directrix is:
[tex]\[
x = -3 - 2 = -5
\][/tex]
### Final Answers
The focus of the given parabola is:
[tex]\[
(-1, -3)
\][/tex]
The directrix of the given parabola is:
[tex]\[
x = -5
\][/tex]
So, the final answers are:
Focus: [tex]\((-1, -3)\)[/tex]
Directrix: [tex]\(x = -5\)[/tex]
