Find the radical axis of co-axial system of circles whose limiting points are (1,2) and (2,3).

Answer

To find the radical axis of the coaxial system of circles whose limiting points are given as (1, 2) and (2, 3), we can follow these steps:

### Step 1: Identify the Points
The limiting points are given as:
- Point A = (1, 2)
- Point B = (2, 3)

### Step 2: Find the Midpoint (P) of AB
The midpoint P of the line segment AB can be calculated using the midpoint formula:
\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ P = \left( \frac{1 + 2}{2}, \frac{2 + 3}{2} \right) = \left( \frac{3}{2}, \frac{5}{2} \right) \]

### Step 3: Calculate the Slope of Line AB
The slope (m) of line AB can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m = \frac{3 - 2}{2 - 1} = \frac{1}{1} = 1 \]

### Step 4: Find the Slope of the Radical Axis
Since the radical axis is the perpendicular bisector of AB, its slope will be the negative reciprocal of the slope of AB:
\[ \text{slope of radical axis} = -\frac{1}{m} = -1 \]

### Step 5: Write the Equation of the Radical Axis
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the midpoint P and \( m \) is the slope of the radical axis:
\[ y - \frac{5}{2} = -1 \left( x - \frac{3}{2} \right) \]
This simplifies to:
\[ y - \frac{5}{2} = -x + \frac{3}{2} \]
Rearranging gives:
\[ y + x = 4 \]

### Step 6: Final Form of the Equation
Rearranging the equation further, we can express it in standard form:
\[ x - y + 1 = 0 \]

### Final Answer
The equation of the radical axis is:
\[ x - y + 1 = 0 \]

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