Family of Circles

Family of Circles:

1. Definition:

  • The family of circles is a set of circles that share some common properties or characteristics. These circles are usually related by certain mathematical rules or transformations.

2. General Equation of a Circle:

  • A circle in the coordinate plane is represented by the equation (xh)2+(yk)2=r2, where (h,k) represents the center and r is the radius.

3. Types of Families of Circles:

  • Concentric Circles: Circles that share the same center but have different radii.
  • Orthogonal Circles: Circles that intersect at right angles.
  • Coaxial Circles: Circles that have the same axes.

4. Properties and Characteristics:

  • Transformation Rules: Circles in a family can be related through translations, rotations, dilations, or combinations of these transformations.
  • Parametric Equations: Some families of circles can be described using parametric equations involving parameters.
  • Geometric Relationships: Circles in a family may exhibit specific geometric relations like tangency, intersection, or sharing common tangents.

5. Equation of a General Family of Circles:

  • The equation S=0 represents a family of circles where S is a second-degree equation in x and y with parameters a, b, and c, typically in the form ax2+by2+2gx+2fy+c=0.
  • By changing the values of a, b, g, f, and c, different circles within the family can be obtained.

Example: 

Consider a family of circles described by the equation 4x2+4y216x8y+k=0, where k is a parameter.

Steps:

  1. Identify Parameters:

    • In the given equation 4x2+4y216x8y+k=0, k is the parameter that influences the family of circles.
  2. Analysis of the Equation:

    • The general equation of a circle is x2+y2+2gx+2fy+c=0. Comparing this with 4x2+4y216x8y+k=0 yields g=4, f=2, and c=k.
  3. Understanding the Family of Circles:

    • The center of the circles within this family can be found using the formulas h=g2 and k=f2. Therefore, the center of these circles is (2,1).
    • The radius r of each circle can be determined using the formula r=h2+k2c

. For this family, the radius is r=5k.

  • Variation with Parameter k:

    • As k changes, the value under the square root changes, influencing the radius of the circles.
    • For k=5, the radius r=55=0. This implies a circle of radius 0, resulting in a single point, i.e., a degenerate circle.
    • For k<5, the family of circles has a positive radius, forming concentric circles expanding from the center (2,1).
    • For k>5, the square root becomes negative, indicating an imaginary radius. Thus, no real circles exist for k>5.
  1. Visual Representation:

    • Plot the circles for different values of k to visualize the changes in their radii and positions within the family.

This example demonstrates how the parameter k in the equation 4x2+4y216x8y+k=0 affects the family of circles' radii and their existence for different values of k. Varying k alters the properties and configurations of circles within the family, influencing their sizes and positions relative to the center (2,1).