Find the length of latus rectum of the parabola `x^(2)=4x-4y`.

Answer

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

### Step 1: Rewrite the equation in standard form
Start with the given equation:
\[ x^2 = 4x - 4y \]
Rearranging the equation gives:
\[ x^2 - 4x = -4y \]

### Step 2: Complete the square
To complete the square on the left side, we take the coefficient of \( x \) (which is -4), halve it to get -2, and then square it to get 4. We add and subtract 4:
\[ x^2 - 4x + 4 - 4 = -4y \]
This simplifies to:
\[ (x - 2)^2 - 4 = -4y \]
Now, add 4 to both sides:
\[ (x - 2)^2 = -4y + 4 \]
This can be rewritten as:
\[ (x - 2)^2 = -4(y - 1) \]

### Step 3: Identify the parameters of the parabola
Now we have the equation in the standard form of a parabola:
\[ (x - h)^2 = 4p(y - k) \]
where \( (h, k) \) is the vertex of the parabola. From our equation, we can see:
- \( h = 2 \)
- \( k = 1 \)
- \( 4p = -4 \) which implies \( p = -1 \)

### Step 4: Determine the length of the latus rectum
The length of the latus rectum of a parabola is given by the formula \( |4p| \). Since \( p = -1 \):
\[ \text{Length of latus rectum} = |4p| = |4 \times -1| = 4 \]

### Final Answer
Thus, the length of the latus rectum of the parabola \( x^2 = 4x - 4y \) is:
\[ \boxed{4} \]

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