The radical centre of the circle `x^(2)+y^(2)=1, x^(2)+y^(2)-2x=1 and x^(2)+y^(2)-2y=1` is

Options

A. (1, 1)
B. (2, 2)
C. (0, 0)
D. none

Answer

To find the radical center of the circles given by the equations \(x^2 + y^2 = 1\), \(x^2 + y^2 - 2x = 1\), and \(x^2 + y^2 - 2y = 1\), we will follow these steps:

### Step 1: Rewrite the Circle Equations
The equations of the circles can be rewritten as:
1. Circle 1: \(S_1: x^2 + y^2 - 1 = 0\)
2. Circle 2: \(S_2: x^2 + y^2 - 2x - 1 = 0\)
3. Circle 3: \(S_3: x^2 + y^2 - 2y - 1 = 0\)

### Step 2: Find the Radical Axis of Circles \(S_1\) and \(S_2\)
The radical axis of two circles can be found using the equation \(S_1 - S_2 = 0\).
\[ S_1 - S_2 = (x^2 + y^2 - 1) - (x^2 + y^2 - 2x - 1) = 2x = 0 \]
Thus, the radical axis of circles \(S_1\) and \(S_2\) is given by:
\[ x = 0 \]

### Step 3: Find the Radical Axis of Circles \(S_2\) and \(S_3\)
Next, we find the radical axis of circles \(S_2\) and \(S_3\):
\[ S_2 - S_3 = (x^2 + y^2 - 2x - 1) - (x^2 + y^2 - 2y - 1) = -2x + 2y = 0 \]
This simplifies to:
\[ y = x \]

### Step 4: Find the Intersection of the Radical Axes
Now, we have two equations:
1. \(x = 0\) (from the radical axis of \(S_1\) and \(S_2\))
2. \(y = x\) (from the radical axis of \(S_2\) and \(S_3\))

Substituting \(x = 0\) into \(y = x\):
\[ y = 0 \]

### Step 5: Conclusion
Thus, the coordinates of the radical center of the three circles are:
\[ (0, 0) \]

### Final Answer
The radical center of the given circles is \((0, 0)\).

---