Coaxial System of Circles
Coaxial System of Circles:
1. Definition:
 Coaxial circles are a set of circles that share the same pair of perpendicular lines as their radical axes.
2. Characteristics:
 Common Radical Axis: All circles in a coaxial system share the same pair of perpendicular lines as their radical axes.
 Equal Power Property: Any point on the common radical axis has equal power with respect to all circles in the system.
3. Properties and Conditions:
 Equation Formulation: For a coaxial system, the equations of the circles can be expressed as $S+\lambda R=0$, where $S$ is a seconddegree equation, $R$ is a firstdegree equation, and $\lambda $ is a parameter representing the family of circles.
 Conditions for Coaxiality: Coaxial circles have the same $S$ but different $R$ in their equation and satisfy $S=0$ and $R=0$ simultaneously.
4. Common Radical Axes:
 The common radical axes of coaxial circles are two perpendicular lines: ${L}_{1}$ and ${L}_{2}$, represented by equations $S=0$ and $R=0$ respectively.
 The equations $S=0$ and $R=0$ intersect at the point of concurrency that is equidistant from all circles in the system.
5. Equation of Coaxial System:
 For a coaxial system of circles with equations ${x}^{2}+{y}^{2}+2gx+2fy+c=0$ and ${x}^{2}+{y}^{2}+2{g}^{\mathrm{\prime}}x+2{f}^{\mathrm{\prime}}y+{c}^{\mathrm{\prime}}=0$, the common radical axes are given by the equation $(g{g}^{\mathrm{\prime}})x+(f{f}^{\mathrm{\prime}})y+(c{c}^{\mathrm{\prime}})=0$.
6. Analyzing Coaxial Systems:
 Geometric Interpretation: Visualizing the positions and relationships between circles with the same radical axes.
 Power of a Point: Any point on the common radical axis has equal power with respect to all circles in the system.
Example: Coaxial System of Circles
Consider two circles ${C}_{1}$ with equation ${x}^{2}+{y}^{2}6x4y+9=0$ and ${C}_{2}$ with equation ${x}^{2}+{y}^{2}8x6y+16=0$.
Steps:

Identifying Centers and Radii:
 Complete the squares for both equations to express them in the standard form $(xh{)}^{2}+(yk{)}^{2}={r}^{2}$ to identify the centers and radii of circles ${C}_{1}$ and ${C}_{2}$.
 For ${C}_{1}$, completing the square yields $(x3{)}^{2}+(y2{)}^{2}=4$ with center $(3,2)$ and radius ${r}_{1}=2$.
 For ${C}_{2}$, completing the square yields $(x4{)}^{2}+(y3{)}^{2}=1$ with center $(4,3)$ and radius ${r}_{2}=1$.

Calculating Radical Axes:
 Use the formula $(g{g}^{\mathrm{\prime}})x+(f{f}^{\mathrm{\prime}})y+(c{c}^{\mathrm{\prime}})=0$ to find the equation of the radical axes. Here, $g,f,c$ and ${g}^{\mathrm{\prime}},{f}^{\mathrm{\prime}},{c}^{\mathrm{\prime}}$are coefficients of ${x}^{2},{y}^{2}$ and constants in the circle equations.
 Substituting values, we get $(68)x+(46)y+(916)=0$ which simplifies to $2x2y7=0$ as the equation of the radical axes.

Interpreting the Result:
 The equation $2x2y7=0$ represents the radical axes of circles ${C}_{1}$ and ${C}_{2}$. These axes are perpendicular to each other and equidistant from the centers of both circles.

Geometric Interpretation:
 The radical axes are lines equidistant from the centers $(3,2)$ and $(4,3)$ of ${C}_{1}$ and ${C}_{2}$ respectively. They intersect at the radical center.