The equation of the circle which cuts the three circles `x^2+y^2-4x-6y+4=0, x^2+y^2-2x-8y+4=0, x^2+y^2-6x-6y+4=0` orthogonally is

Options

A. `x^2+y^2=4`
B. `x^2+y^2=2`
C. `x^2+y^2=1`
D. `x^2+y^2=8`

Answer

To find the equation of the circle that cuts the three given circles orthogonally, we follow these steps:

### Step 1: Write the equations of the given circles
The equations of the three circles are:
1. \( S_1: x^2 + y^2 - 4x - 6y + 4 = 0 \)
2. \( S_2: x^2 + y^2 - 2x - 8y + 4 = 0 \)
3. \( S_3: x^2 + y^2 - 6x - 6y + 4 = 0 \)

### Step 2: Find the radical axis of the first two circles
The radical axis between two circles \( S_1 \) and \( S_2 \) is found by subtracting their equations:
\[ S_1 - S_2 = 0 \]
Calculating this gives:
\[ (x^2 + y^2 - 4x - 6y + 4) - (x^2 + y^2 - 2x - 8y + 4) = 0 \]
Cancelling \( x^2 \) and \( y^2 \):
\[ -4x - 6y + 4 + 2x + 8y - 4 = 0 \]
Simplifying:
\[ -2x + 2y = 0 \quad \Rightarrow \quad x - y = 0 \quad \Rightarrow \quad x = y \]

### Step 3: Find the radical axis of the second and third circles
Now, we find the radical axis between \( S_2 \) and \( S_3 \):
\[ S_2 - S_3 = 0 \]
Calculating this gives:
\[ (x^2 + y^2 - 2x - 8y + 4) - (x^2 + y^2 - 6x - 6y + 4) = 0 \]
Cancelling \( x^2 \) and \( y^2 \):
\[ -2x - 8y + 4 + 6x + 6y - 4 = 0 \]
Simplifying:
\[ 4x - 2y = 0 \quad \Rightarrow \quad 2x - y = 0 \quad \Rightarrow \quad y = 2x \]

### Step 4: Find the intersection of the two radical axes
We have the equations:
1. \( x = y \)
2. \( y = 2x \)

Substituting \( y = x \) into \( y = 2x \):
\[ x = 2x \quad \Rightarrow \quad x = 0 \quad \Rightarrow \quad y = 0 \]
Thus, the radical center is at \( (0, 0) \).

### Step 5: Find the radius of the required circle
The radius of the required circle can be found by calculating the distance from the radical center \( (0, 0) \) to any of the given circles. We can choose \( S_1 \):
\[ S_1: 0^2 + 0^2 - 4(0) - 6(0) + 4 = 4 \]
The radius \( r \) is given by:
\[ r = \sqrt{4} = 2 \]

### Step 6: Write the equation of the required circle
Using the center-radius form of the circle:
\[ (x - 0)^2 + (y - 0)^2 = 2^2 \]
This simplifies to:
\[ x^2 + y^2 = 4 \]

### Final Answer
The equation of the circle that cuts the three given circles orthogonally is:
\[ \boxed{x^2 + y^2 = 4} \]