The focus of the parabola `y ^(2) =4y -4x` is

Options

A. `(0,2)`
B. `(1,2)`
C. `(2,0)`
D. `(2,1)`

Answer

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

### Step 1: Rearrange the equation
Start with the given equation:
\[ y^2 = 4y - 4x \]
Rearranging it, we can write:
\[ y^2 - 4y = -4x \]

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

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

From our equation, we can identify:
- \(h = 1\)
- \(k = 2\)
- \(4a = 4 \Rightarrow a = 1\)

### Step 4: Find the vertex
The vertex of the parabola is at the point \((h, k) = (1, 2)\).

### Step 5: Determine the focus
For a parabola that opens to the left (as indicated by the negative sign), the focus is located \(a\) units to the left of the vertex. Since \(a = 1\), we move 1 unit left from the vertex:
\[ \text{Focus} = (h - a, k) = (1 - 1, 2) = (0, 2) \]

### Final Answer
Thus, the focus of the parabola \(y^2 = 4y - 4x\) is:
\[ \boxed{(0, 2)} \]