The equation of the directrix of the parabola `x^(2) - 4x - 8y + 12 = 0 ` is

Options

A. `y = 0`
B. `x - 1 = 0`
C. `y+ 1 = 0`
D. `x + 1 = 0`

Answer

To find the equation of the directrix of the parabola given by the equation \(x^2 - 4x - 8y + 12 = 0\), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation into a more standard form. We can rewrite the equation as:
\[ x^2 - 4x = 8y - 12 \]

### Step 2: Complete the square
Next, we complete the square for the \(x\) terms on the left side:
\[ x^2 - 4x + 4 = 8y - 12 + 4 \]
This simplifies to:
\[ (x - 2)^2 = 8y - 8 \]
or
\[ (x - 2)^2 = 8(y - 1) \]

### Step 3: Identify parameters
Now, we can compare this equation with the standard form of a parabola that opens upwards, which is:
\[ (x - a)^2 = 4b(y - c) \]
From our equation, we can identify:
- \(a = 2\)
- \(4b = 8\) which gives \(b = 2\)
- \(c = 1\)

### Step 4: Find the directrix
The directrix of a parabola in this form is given by the equation:
\[ y - c = -b \]
Substituting the values we found:
\[ y - 1 = -2 \]
This simplifies to:
\[ y = -1 \]

### Conclusion
Thus, the equation of the directrix of the parabola is:
\[ y + 1 = 0 \]