Radical Axis of Circles
Radical Axis of Circles:
1. Definition:
 The radical axis of two circles is a line that is perpendicular to the line joining their centers and whose power of a point is the same with respect to both circles.
2. Properties:
 Perpendicularity: The radical axis is perpendicular to the line joining the centers of the circles.
 Power of a Point: The power of any point on the radical axis with respect to both circles is equal.
3. Equation of the Radical Axis:
 The radical axis of two circles ${C}_{1}$ and ${C}_{2}$ with centers $({h}_{1},{k}_{1})$ and $({h}_{2},{k}_{2})$ and radii ${r}_{1}$ and ${r}_{2}$ respectively, is given by the equation: $(x{x}_{0})({h}_{1}{h}_{2})+(y{y}_{0})({k}_{1}{k}_{2})=\frac{1}{2}({r}_{1}^{2}{r}_{2}^{2})$.
 The midpoint $({x}_{0},{y}_{0})$ of the line joining the centers is the point of intersection of the radical axis and the line joining the centers of the circles.
4. Analyzing the Radical Axis:
 Power of a Point: Any point on the radical axis has the same power with respect to both circles.
 Geometric Interpretation: Helps in determining the common tangents, intersecting points, or relative positions of circles.
5. Applications:
 Circle Geometry: Useful in analyzing the relationships and configurations of circles in the coordinate plane.
 Geometry Problem Solving: Helps in solving problems involving circles and their properties.
Example: Radical Axis of Circles
Consider two circles ${C}_{1}$ with center $(3,2)$ and radius $4$ and ${C}_{2}$ with center $(1,6)$ and radius $3$.
Steps:

Finding Equation of the Radical Axis:
 Determine the equation of the line perpendicular to the line joining the centers and passing through the midpoint of the segment joining the centers of ${C}_{1}$ and ${C}_{2}$.

Calculating Midpoint of the Centers:
 The midpoint of the segment joining the centers is
Example: Radical Axis of Circles
Consider two circles ${C}_{1}$ with center $(3,2)$ and radius $4$ and ${C}_{2}$ with center $(1,6)$ and radius $3$.
Steps:

Finding Equation of the Radical Axis:
 Determine the equation of the line perpendicular to the line joining the centers and passing through the midpoint of the segment joining the centers of ${C}_{1}$ and ${C}_{2}$.

Calculating Midpoint of the Centers:
 The midpoint of the segment joining the centers is $(\frac{31}{2},\frac{2+6}{2})=(1,4)$.

Calculating Radical Axis Equation:
 The equation of the line passing through $(1,4)$ and perpendicular to the line joining the centers is perpendicular to the line joining the centers and passes through the midpoint.
 The slope of the line joining the centers is $\frac{62}{13}=\frac{4}{4}=1$. Hence, the slope of the radical axis line is $1$ (negative reciprocal).
 Therefore, the equation of the radical axis is $y4=1\cdot (x1)$ or $y=x+3$.

Interpreting the Result:
 The equation $y=x+3$ represents the radical axis of the two circles. Any point on this line has equal power with respect to both circles.

Geometric Interpretation:
 The radical axis $y=x+3$ is a line passing through $(1,4)$ and perpendicular to the line joining the centers. It is equidistant from the circles ${C}_{1}$ and ${C}_{2}$.
This example demonstrates how to determine the radical axis of two circles by finding the equation of the line perpendicular to the line joining the centers and passing through the midpoint. The radical axis plays a crucial role in understanding the relationship between circles and their common properties.

 The midpoint of the segment joining the centers is

Calculating Radical Axis Equation:
 The equation of the line passing through $(1,4)$ and perpendicular to the line joining the centers is perpendicular to the line joining the centers and passes through the midpoint.
 The slope of the line joining the centers is $\frac{62}{13}=\frac{4}{4}=1$. Hence, the slope of the radical axis line is $1$ (negative reciprocal).
 Therefore, the equation of the radical axis is $y4=1\cdot (x1)$ or $y=x+3$.

Interpreting the Result:
 The equation $y=x+3$ represents the radical axis of the two circles. Any point on this line has equal power with respect to both circles.

Geometric Interpretation:
 The radical axis $y=x+3$ is a line passing through $(1,4)$ and perpendicular to the line joining the centers. It is equidistant from the circles ${C}_{1}$ and ${C}_{2}$.