Find the coordinates of the centres and the radii of the circles whose equations are
`x^(2) + y^(2) = 2gx - 2fy`

Answer

To find the coordinates of the center and the radius of the circle given by the equation \( x^2 + y^2 = 2gx - 2fy \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x^2 + y^2 = 2gx - 2fy \]
We can rearrange it to bring all terms to one side:
\[ x^2 - 2gx + y^2 + 2fy = 0 \]

### Step 2: Completing the Square
Next, we will complete the square for the \(x\) and \(y\) terms.

For the \(x\) terms:
\[ x^2 - 2gx = (x - g)^2 - g^2 \]

For the \(y\) terms:
\[ y^2 + 2fy = (y + f)^2 - f^2 \]

Now, substituting these completed squares back into the equation gives:
\[ (x - g)^2 - g^2 + (y + f)^2 - f^2 = 0 \]

### Step 3: Simplifying the Equation
Now, we can simplify the equation:
\[ (x - g)^2 + (y + f)^2 - (g^2 + f^2) = 0 \]
This can be rewritten as:
\[ (x - g)^2 + (y + f)^2 = g^2 + f^2 \]

### Step 4: Identifying the Center and Radius
From the standard form of the circle equation \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- The center \((h, k)\) is \((g, -f)\).
- The radius \(r\) is \(\sqrt{g^2 + f^2}\).

### Final Answer
Thus, the coordinates of the center of the circle are \((g, -f)\) and the radius is \(\sqrt{g^2 + f^2}\).