The vertex of the parabola `x^2 + y^2 - 2xy - 4x - 4y + 4 = 0` is at

Options

A. (1, 1)
B. (-1, -1)
C. `(1/2,1/2)`
D. none of these

Explanation

To find the vertex of the parabola given by the equation \(x^2 + y^2 - 2xy - 4x - 4y + 4 = 0\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ x^2 + y^2 - 2xy - 4x - 4y + 4 = 0 \]

### Step 2: Rearrange the equation
Rearranging the equation, we can group the terms:
\[ x^2 - 2xy + y^2 - 4x - 4y + 4 = 0 \]

### Step 3: Complete the square
We can rewrite \(x^2 - 2xy + y^2\) as \((x - y)^2\):
\[ (x - y)^2 - 4x - 4y + 4 = 0 \]

Next, we can rearrange the remaining terms:
\[ (x - y)^2 = 4x + 4y - 4 \]

### Step 4: Factor out common terms
Factoring out a 4 from the right side:
\[ (x - y)^2 = 4(x + y - 1) \]

### Step 5: Divide by 2
To convert it into a standard form, we can divide both sides by 2:
\[ \frac{(x - y)^2}{2} = 2(x + y - 1) \]

### Step 6: Rewrite in standard form
This can be rewritten in a more recognizable form:
\[ \left(\frac{x - y}{\sqrt{2}}\right)^2 = 2(x + y - 1) \]

### Step 7: Identify the vertex
The vertex of the parabola in the form \(y^2 = 4px\) can be identified. Here, we need to find the point where the parabola opens. Set \(x + y - 1 = 0\) and \(x - y = 0\) to find the intersection point.

From \(x - y = 0\), we have:
\[ x = y \]

Substituting \(y\) with \(x\) in \(x + y - 1 = 0\):
\[ x + x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]

Since \(x = y\), we also have:
\[ y = \frac{1}{2} \]

### Final Answer
Thus, the vertex of the parabola is:
\[ \left(\frac{1}{2}, \frac{1}{2}\right) \]

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