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 $(xh{)}^{2}+(yk{)}^{2}={r}^{2}$, 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 seconddegree equation in $x$ and $y$ with parameters $a$, $b$, and $c$, typically in the form $a{x}^{2}+b{y}^{2}+2gx+2fy+c=0$.
 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 $4{x}^{2}+4{y}^{2}16x8y+k=0$, where $k$ is a parameter.
Steps:

Identify Parameters:
 In the given equation $4{x}^{2}+4{y}^{2}16x8y+k=0$, $k$ is the parameter that influences the family of circles.

Analysis of the Equation:
 The general equation of a circle is ${x}^{2}+{y}^{2}+2gx+2fy+c=0$. Comparing this with $4{x}^{2}+4{y}^{2}16x8y+k=0$ yields $g=4$, $f=2$, and $c=k$.

Understanding the Family of Circles:
 The center of the circles within this family can be found using the formulas $h=\frac{g}{2}$ and $k=\frac{f}{2}$. Therefore, the center of these circles is $(2,1)$.
 The radius $r$ of each circle can be determined using the formula $r=\sqrt{{h}^{2}+{k}^{2}c}$
. For this family, the radius is $r=\sqrt{5k}$

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 $r=\sqrt{55}=0$

 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 $k>5$.

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 $4{x}^{2}+4{y}^{2}16x8y+k=0$ 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 $(2,1)$.