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+λR=0, where S is a second-degree equation, R is a first-degree equation, and λ 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: L1 and L2, 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 x2+y2+2gx+2fy+c=0 and x2+y2+2gx+2fy+c=0, the common radical axes are given by the equation (gg)x+(ff)y+(cc)=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 C1 with equation x2+y26x4y+9=0 and C2 with equation x2+y28x6y+16=0.

Steps:

  1. Identifying Centers and Radii:

    • Complete the squares for both equations to express them in the standard form (xh)2+(yk)2=r2 to identify the centers and radii of circles C1 and C2.
    • For C1, completing the square yields (x3)2+(y2)2=4 with center (3,2) and radius r1=2.
    • For C2, completing the square yields (x4)2+(y3)2=1 with center (4,3) and radius r2=1.
  2. Calculating Radical Axes:

    • Use the formula (gg)x+(ff)y+(cc)=0 to find the equation of the radical axes. Here, g,f,c and g,f,care coefficients of x2,y2 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.
  3. Interpreting the Result:

    • The equation 2x2y7=0 represents the radical axes of circles C1 and C2. These axes are perpendicular to each other and equidistant from the centers of both circles.
  4. Geometric Interpretation:

    • The radical axes are lines equidistant from the centers (3,2) and (4,3) of C1 and C2 respectively. They intersect at the radical center.