Family of Circles - Passing Through Point of Intersection of Two Circles

The family of circles passing through the points of intersection of two given circles is an important concept in coordinate geometry, particularly in the study of circles. This family is a set of all possible circles that pass through the common points of two intersecting circles.

Understanding the Concept

When two circles intersect, they do so at two points (assuming the circles are not tangent to each other). Any circle that passes through these two points is said to be a member of the family of circles defined by the original pair of intersecting circles.

General Equation of a Circle

The general equation of a circle in the Cartesian plane is given by:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

where $(h, k)$ is the center of the circle and $r$ is the radius.

Equation of the Family of Circles

The equation of the family of circles passing through the intersection points of two circles with equations $C_1: x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $C_2: x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ can be represented as:

$$ C: x^2 + y^2 + 2gx + 2fy + c = 0 $$

where $g = g_1 + \lambda g_2$, $f = f_1 + \lambda f_2$, and $c = c_1 + \lambda c_2$ for some parameter $\lambda$.

Derivation

The family of circles can be derived using the concept of linear combination. If $P(x, y)$ is the point of intersection of the two given circles $C_1$ and $C_2$, then $P$ satisfies both equations. For any value of $\lambda$, the equation $C_1 + \lambda C_2 = 0$ will also be satisfied by $P$. This equation represents the entire family of circles passing through the intersection points of $C_1$ and $C_2$.

Table of Differences and Important Points

Feature	Circle $C_1$	Circle $C_2$	Family of Circles $C$
Equation	$x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$	$x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$	$x^2 + y^2 + 2gx + 2fy + c = 0$
Center	$(g_1, f_1)$	$(g_2, f_2)$	$(g, f)$ where $g = g_1 + \lambda g_2$ and $f = f_1 + \lambda f_2$
Radius	$\sqrt{g_1^2 + f_1^2 - c_1}$	$\sqrt{g_2^2 + f_2^2 - c_2}$	Varies with $\lambda$
Parameter	Not applicable	Not applicable	$\lambda$ determines the specific member of the family

Examples

Example 1: Finding the Family of Circles

Given two circles $C_1: x^2 + y^2 - 4x - 6y + 9 = 0$ and $C_2: x^2 + y^2 + 2x - 2y = 0$, find the equation of the family of circles passing through their points of intersection.

Solution:

The family of circles can be represented as:

$$ C: (x^2 + y^2 - 4x - 6y + 9) + \lambda(x^2 + y^2 + 2x - 2y) = 0 $$

Simplifying, we get:

$$ C: x^2 + y^2 + 2(-2 + \lambda)x + 2(-3 - \lambda)y + (9 + \lambda) = 0 $$

This is the general equation of the family of circles for any value of $\lambda$.

Example 2: Specific Member of the Family

Find a specific member of the family of circles from Example 1 that has its center on the x-axis.

Solution:

For the center to be on the x-axis, the y-coordinate of the center must be zero. This implies $f = -3 - \lambda = 0$, so $\lambda = -3$.

Substituting $\lambda = -3$ into the family equation, we get:

$$ C: x^2 + y^2 + 2(1)x + 6 = 0 $$

This is the equation of the specific circle in the family with its center on the x-axis.

Conclusion

The family of circles passing through the points of intersection of two given circles is a fundamental concept in circle geometry. By understanding the derivation and properties of this family, one can solve a variety of problems related to circle configurations and their intersections. The parameter $\lambda$ plays a crucial role in defining the specific members of this family.