For the parabola `y^(2)+8x-12y+20=0`

Options

A. Vertex is (2,6)
B. Focus is (0,6)
C. Latursrectum 4
D. All of these

Explanation

To solve the parabola given by the equation \( y^2 + 8x - 12y + 20 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the \( y^2 \) and \( y \) terms on one side:
\[ y^2 - 12y + 8x + 20 = 0 \]

### Step 2: Complete the square for the \( y \) terms
To complete the square for the \( y \) terms, we take the coefficient of \( y \), which is -12, halve it to get -6, and square it to get 36. We will add and subtract 36:
\[ y^2 - 12y + 36 - 36 + 8x + 20 = 0 \]
This simplifies to:
\[ (y - 6)^2 - 16 + 8x = 0 \]

### Step 3: Simplify the equation
Now, we can simplify the equation:
\[ (y - 6)^2 = -8x + 16 \]
Rearranging gives:
\[ (y - 6)^2 = 16 - 8x \]

### Step 4: Write in standard form
We can rewrite this in the standard form of a parabola:
\[ (y - 6)^2 = -8(x - 2) \]
This shows that the parabola opens to the left with vertex at \( (2, 6) \).

### Step 5: Identify the vertex
From the standard form, we can identify the vertex:
\[ \text{Vertex} = (2, 6) \]

### Step 6: Identify the focus
For a parabola in the form \( (y - k)^2 = 4p(x - h) \), the focus is located at \( (h + p, k) \). Here, \( h = 2 \), \( k = 6 \), and \( 4p = -8 \) implies \( p = -2 \). Thus, the focus is:
\[ \text{Focus} = (2 - 2, 6) = (0, 6) \]

### Step 7: Find the equation of the directrix
The directrix of the parabola is given by the equation \( x = h - p \):
\[ \text{Directrix} = x = 2 + 2 = 4 \]

### Step 8: Determine the length of the latus rectum
The length of the latus rectum is given by \( |4p| \):
\[ \text{Length of latus rectum} = |4 \times -2| = 8 \]

### Summary of Results
- Vertex: \( (2, 6) \)
- Focus: \( (0, 6) \)
- Directrix: \( x = 4 \)
- Length of latus rectum: \( 8 \)