To solve the equation of the parabola given by \( y^2 + 8x - 12y + 20 = 0 \), we will follow these steps:
### Step 1: Rearranging the Equation
We start with the given equation:
\[
y^2 + 8x - 12y + 20 = 0
\]
Rearranging it, we isolate the \( y \) terms on one side:
\[
y^2 - 12y + 8x + 20 = 0
\]
### Step 2: Completing the Square
Next, we complete the square for the \( y \) terms:
\[
y^2 - 12y = (y - 6)^2 - 36
\]
Substituting this back into the equation gives:
\[
(y - 6)^2 - 36 + 8x + 20 = 0
\]
Simplifying this, we have:
\[
(y - 6)^2 + 8x - 16 = 0
\]
or
\[
(y - 6)^2 = -8x + 16
\]
which can be rewritten as:
\[
(y - 6)^2 = -8(x - 2)
\]
### Step 3: Identifying the Parabola's Characteristics
From the equation \( (y - 6)^2 = -8(x - 2) \), we can identify the following:
- This is a parabola that opens to the left.
- The vertex of the parabola is at \( (2, 6) \).
### Step 4: Finding the Focus
For a parabola of the form \( (y - k)^2 = -4a(x - h) \), the focus is located at \( (h - a, k) \).
Here, \( h = 2 \), \( k = 6 \), and \( 4a = 8 \) implies \( a = 2 \).
Thus, the coordinates of the focus are:
\[
(2 - 2, 6) = (0, 6)
\]
### Step 5: Finding the Length of the Latus Rectum
The length of the latus rectum for a parabola is given by \( 4a \). Since \( a = 2 \):
\[
\text{Length of the Latus Rectum} = 4 \times 2 = 8
\]
### Step 6: Finding the Axis of the Parabola
The axis of the parabola is the line \( y = k \). Here, since \( k = 6 \):
\[
\text{Axis: } y = 6
\]
### Summary of Results
- **Vertex**: \( (2, 6) \)
- **Focus**: \( (0, 6) \)
- **Length of Latus Rectum**: \( 8 \)
- **Axis**: \( y = 6 \)
### Conclusion
Now we can check the options provided:
1. Vertex \( (2, 6) \) - True
2. Focus \( (0, 6) \) - True
3. Axis \( y = 6 \) - True
4. Length of Latus Rectum \( 8 \) - True
