# 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 second-degree equation, $R$ is a first-degree 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.

• 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 Co-axial System:

• For a co-axial 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 $\left(g-{g}^{\mathrm{\prime }}\right)x+\left(f-{f}^{\mathrm{\prime }}\right)y+\left(c-{c}^{\mathrm{\prime }}\right)=0$.

#### 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: Co-axial System of Circles

Consider two circles ${C}_{1}$ with equation ${x}^{2}+{y}^{2}-6x-4y+9=0$ and ${C}_{2}$ with equation ${x}^{2}+{y}^{2}-8x-6y+16=0$.

#### Steps:

• Complete the squares for both equations to express them in the standard form $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$ to identify the centers and radii of circles ${C}_{1}$ and ${C}_{2}$.
• For ${C}_{1}$, completing the square yields $\left(x-3{\right)}^{2}+\left(y-2{\right)}^{2}=4$ with center $\left(3,2\right)$ and radius ${r}_{1}=2$.
• For ${C}_{2}$, completing the square yields $\left(x-4{\right)}^{2}+\left(y-3{\right)}^{2}=1$ with center $\left(4,3\right)$ and radius ${r}_{2}=1$.
• Use the formula $\left(g-{g}^{\mathrm{\prime }}\right)x+\left(f-{f}^{\mathrm{\prime }}\right)y+\left(c-{c}^{\mathrm{\prime }}\right)=0$ 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 $\left(6-8\right)x+\left(4-6\right)y+\left(9-16\right)=0$ which simplifies to $-2x-2y-7=0$ as the equation of the radical axes.
• The equation $-2x-2y-7=0$ 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.
• The radical axes are lines equidistant from the centers $\left(3,2\right)$ and $\left(4,3\right)$ of ${C}_{1}$ and ${C}_{2}$ respectively. They intersect at the radical center.