The circle orthogonal to the three circles `x^2+y^2+a_i x +b_i y+c=0`, (i=1,2,3) is

Options

A. `x^2+y^2-b_i x-a_i y-c=0`
B. `x^2+y^2=c`
C. `x^2+y^2=a_i + b_i`
D. `x^2+y^2=a_i`

Answer

To find the circle orthogonal to the three given circles \(x^2 + y^2 + a_i x + b_i y + c = 0\) for \(i = 1, 2, 3\), we can follow these steps:

### Step 1: Write the equations of the circles
The equations of the three circles can be represented as:
1. \(S_1 = x^2 + y^2 + a_1 x + b_1 y + c = 0\)
2. \(S_2 = x^2 + y^2 + a_2 x + b_2 y + c = 0\)
3. \(S_3 = x^2 + y^2 + a_3 x + b_3 y + c = 0\)

### Step 2: Use the orthogonality condition
For two circles to be orthogonal, the condition is given by:
\[ S_1 S_2 = 0 \]
This means we can find the equations for the differences:
- \(S_1 - S_2 = 0\)
- \(S_2 - S_3 = 0\)

### Step 3: Set up the equations
From the orthogonality condition, we can write:
1. \(S_1 - S_2 = (x^2 + y^2 + a_1 x + b_1 y + c) - (x^2 + y^2 + a_2 x + b_2 y + c) = 0\)
- This simplifies to:
\[ (a_1 - a_2)x + (b_1 - b_2)y = 0 \]
2. Similarly for \(S_2 - S_3\):
\[ (a_2 - a_3)x + (b_2 - b_3)y = 0 \]

### Step 4: Solve the equations
From the two equations derived, we can conclude:
- If both equations hold true, then \(x\) and \(y\) must be equal to zero:
\[ x = 0, \quad y = 0 \]

### Step 5: Find the radius of the orthogonal circle
The center of the orthogonal circle is at the origin \((0, 0)\). To find the radius \(r\), we use the formula:
\[ r = \sqrt{0^2 + 0^2 + c} = \sqrt{c} \]

### Step 6: Write the equation of the orthogonal circle
The equation of the circle with center at \((0, 0)\) and radius \(r\) is given by:
\[ x^2 + y^2 = r^2 \]
Substituting for \(r\):
\[ x^2 + y^2 = c \]

### Conclusion
Thus, the circle orthogonal to the three given circles is:
\[ x^2 + y^2 = c \]