The length of latus - rectum of the parabola `x^(2) - 4x + 8y + 12 = 0 ` is (i) 2 (ii) 4 (iii) 6 (iv) 8

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the length of the latus rectum 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 standard form. The given equation is:
\[ x^2 - 4x + 8y + 12 = 0 \]
We can rewrite it as:
\[ x^2 - 4x = -8y - 12 \]

### Step 2: Complete the square
Next, we complete the square for the \(x\) terms. We take the expression \(x^2 - 4x\) and complete the square:
\[ x^2 - 4x + 4 - 4 = (x - 2)^2 - 4 \]
Substituting this back into the equation gives:
\[ (x - 2)^2 - 4 = -8y - 12 \]
Now, we can rearrange it:
\[ (x - 2)^2 = -8y - 12 + 4 \]
\[ (x - 2)^2 = -8y - 8 \]
\[ (x - 2)^2 = -8(y + 1) \]

### Step 3: Identify the standard form
Now, we can compare this with the standard form of a parabola:
\[ (x - h)^2 = 4b(y - k) \]
From our equation \((x - 2)^2 = -8(y + 1)\), we can identify:
- \(h = 2\)
- \(k = -1\)
- \(4b = -8\)

### Step 4: Solve for \(b\)
From \(4b = -8\), we can solve for \(b\):
\[ b = \frac{-8}{4} = -2 \]
Since \(b\) represents the distance, we take the absolute value:
\[ b = 2 \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of a parabola is given by the formula:
\[ L = 4b \]
Substituting the value of \(b\):
\[ L = 4 \times 2 = 8 \]

### Conclusion
Thus, the length of the latus rectum of the given parabola is:
\[ \boxed{8} \]