The radical axis of the circles, belonging to the coaxal system of circles whose limiting points are (1, 3) and (2, 6), is

Options

A. `x-3y-15=0`
B. `x+3y-15=0`
C. `x-3y+15=0`
D. `2x+3y-15=0`

Explanation

To find the radical axis of the circles belonging to the coaxial system with limiting points (1, 3) and (2, 6), we can follow these steps:

### Step 1: Identify the Limiting Points
The limiting points of the coaxial circles are given as (1, 3) and (2, 6). These points serve as the centers of the circles in the coaxial system.

### Step 2: Write the Equations of the Circles
Since the radius of the circles at the limiting points is zero, the equations of the circles can be written as:
- For Circle 1 (center at (1, 3)):
\[ S_1: (x - 1)^2 + (y - 3)^2 = 0 \]
- For Circle 2 (center at (2, 6)):
\[ S_2: (x - 2)^2 + (y - 6)^2 = 0 \]

### Step 3: Set Up the Equation for the Radical Axis
The equation of the radical axis is given by:
\[ S_1 - S_2 = 0 \]
Substituting the equations of the circles:
\[ [(x - 1)^2 + (y - 3)^2] - [(x - 2)^2 + (y - 6)^2] = 0 \]

### Step 4: Expand the Equations
Now, we will expand both sides:
- Expanding \(S_1\):
\[ (x - 1)^2 + (y - 3)^2 = x^2 - 2x + 1 + y^2 - 6y + 9 = x^2 + y^2 - 2x - 6y + 10 \]
- Expanding \(S_2\):
\[ (x - 2)^2 + (y - 6)^2 = x^2 - 4x + 4 + y^2 - 12y + 36 = x^2 + y^2 - 4x - 12y + 40 \]

### Step 5: Substitute and Simplify
Now substituting back into the radical axis equation:
\[ [x^2 + y^2 - 2x - 6y + 10] - [x^2 + y^2 - 4x - 12y + 40] = 0 \]
This simplifies to:
\[ -2x - 6y + 10 + 4x + 12y - 40 = 0 \]
Combining like terms:
\[ (4x - 2x) + (-6y + 12y) + (10 - 40) = 0 \]
This results in:
\[ 2x + 6y - 30 = 0 \]

### Step 6: Finalize the Equation
Dividing the entire equation by 2 gives:
\[ x + 3y - 15 = 0 \]

### Conclusion
The equation of the radical axis is:
\[ x + 3y - 15 = 0 \]