The length of latus rectum of the parabola `x^2 - 4x -8y +12=0` is :

Options

A. `4`
B. `6`
C. `8`
D. `10`

Answer

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

### Step 1: Rearrange the equation into standard form
Start with the given equation:
\[ x^2 - 4x - 8y + 12 = 0 \]
Rearranging gives:
\[ x^2 - 4x = 8y - 12 \]

### Step 2: Complete the square for the \(x\) terms
To complete the square for the left side:
\[ x^2 - 4x = (x - 2)^2 - 4 \]
Substituting this back into the equation gives:
\[ (x - 2)^2 - 4 = 8y - 12 \]
Adding 4 to both sides results in:
\[ (x - 2)^2 = 8y - 8 \]
Or:
\[ (x - 2)^2 = 8(y - 1) \]

### Step 3: Identify the standard form of the parabola
The equation \((x - 2)^2 = 8(y - 1)\) is now in the standard form of a parabola:
\[ (x - h)^2 = 4a(y - k) \]
where \(h = 2\), \(k = 1\), and \(4a = 8\).

### Step 4: Solve for \(a\)
From \(4a = 8\), we can find:
\[ a = \frac{8}{4} = 2 \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum of a parabola is given by the formula \(4a\). Thus:
\[ \text{Length of latus rectum} = 4a = 4 \times 2 = 8 \]

### Conclusion
The length of the latus rectum of the parabola \(x^2 - 4x - 8y + 12 = 0\) is \(8\).

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