Find the radical axis of a co-axial system of circles whose limiting points are (1,2) and (3,4).

Answer

To find the radical axis of a coaxial system of circles with limiting points (1, 2) and (3, 4), we will follow these steps:

### Step 1: Find the Midpoint of the Limiting Points
The midpoint \( P \) of the points \( (1, 2) \) and \( (3, 4) \) can be calculated using the midpoint formula:
\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the values:
\[ P = \left( \frac{1 + 3}{2}, \frac{2 + 4}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \]

### Step 2: Find the Slope of the Line Connecting the Limiting Points
The slope \( m_1 \) of the line connecting the points \( (1, 2) \) and \( (3, 4) \) is given by:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1 \]

### Step 3: Find the Slope of the Perpendicular Line
The slope \( m_2 \) of the line perpendicular to the line connecting the limiting points can be found using the relationship:
\[ m_1 \cdot m_2 = -1 \]
Thus,
\[ m_2 = -\frac{1}{m_1} = -1 \]

### Step 4: Write the Equation of the Radical Axis
Using the point-slope form of the equation of a line, we can write the equation of the radical axis:
\[ y - y_1 = m_2 (x - x_1) \]
Substituting the midpoint \( P(2, 3) \) and the slope \( m_2 = -1 \):
\[ y - 3 = -1(x - 2) \]
Simplifying this equation:
\[ y - 3 = -x + 2 \]
\[ y + x - 5 = 0 \]
Thus, the equation of the radical axis is:
\[ x + y - 5 = 0 \]

### Summary of the Solution
The radical axis of the coaxial system of circles with limiting points (1, 2) and (3, 4) is given by the equation:
\[ x + y - 5 = 0 \]