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 $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$, where $\left(h,k\right)$ 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 $a{x}^{2}+b{y}^{2}+2gx+2fy+c=0$.
• 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 $4{x}^{2}+4{y}^{2}-16x-8y+k=0$, where $k$ is a parameter.

Steps:

1. Identify Parameters:

• In the given equation $4{x}^{2}+4{y}^{2}-16x-8y+k=0$, $k$ is the parameter that influences the family of circles.
2. Analysis of the Equation:

• The general equation of a circle is ${x}^{2}+{y}^{2}+2gx+2fy+c=0$. Comparing this with $4{x}^{2}+4{y}^{2}-16x-8y+k=0$ 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 $h=-\frac{g}{2}$ and $k=-\frac{f}{2}$. Therefore, the center of these circles is $\left(2,1\right)$.
• 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 $r=\sqrt{5-k}$.

• 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 $r=\sqrt{5-5}=0$. 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 $\left(2,1\right)$.
• 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 $4{x}^{2}+4{y}^{2}-16x-8y+k=0$ 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 $\left(2,1\right)$.