Family of Circles - Passing Through Point of Intersection of Line and a Circle

The concept of the family of circles is an important topic in coordinate geometry, particularly in the study of circles. A family of circles refers to a set of circles that share a common geometric property. In this context, we will explore the family of circles that pass through the points of intersection of a given line and a circle.

Understanding the Intersection of a Line and a Circle

Before diving into the family of circles, let's understand the intersection of a line and a circle. A line can intersect a circle at two points, one point (tangent), or not at all. The points of intersection are found by solving the system of equations representing the line and the circle simultaneously.

Equation of a Circle

The general equation of a circle with center at $(h, k)$ and radius $r$ is given by:

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

Equation of a Line

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

$$ Ax + By + C = 0 $$

Points of Intersection

To find the points of intersection, we solve the equations of the line and the circle together. The solutions to this system of equations give us the coordinates of the intersection points.

Family of Circles Passing Through Intersection Points

When a line intersects a circle, it creates two points of intersection (assuming the line is not a tangent or does not miss the circle). There is an infinite number of circles that can pass through these two points. These circles constitute a family of circles.

General Equation of the Family of Circles

The general equation of the family of circles that pass through the intersection points of a given line $L: Ax + By + C = 0$ and a given circle $S: (x - h)^2 + (y - k)^2 = r^2$ is given by:

$$ S + \lambda L = 0 $$

where $\lambda$ is a parameter.

This equation represents the family of circles because for each value of $\lambda$, we get a different circle that passes through the intersection points.

Important Points and Differences

Here is a table summarizing the important points and differences:

Aspect	Line-Circle Intersection	Family of Circles
Definition	The points where a line meets a circle.	A set of circles that pass through the same two points.
Equation	Solve $Ax + By + C = 0$ and $(x - h)^2 + (y - k)^2 = r^2$ together.	$S + \lambda L = 0$ where $S$ is the circle equation and $L$ is the line equation.
Parameter	Not applicable.	$\lambda$ is a real number that varies to generate different circles in the family.
Tangent Case	If the line is tangent to the circle, there is only one point of intersection.	The family of circles includes the given circle as a special case when $\lambda = 0$.
No Intersection	If the line does not intersect the circle, there are no real points of intersection.	Not applicable, as there must be two points of intersection to define the family.

Examples

Example 1: Finding the Family of Circles

Given a circle $S: (x - 2)^2 + (y - 3)^2 = 4$ and a line $L: x + y - 5 = 0$, find the equation of the family of circles passing through their points of intersection.

Solution:

The family of circles is given by:

$$ (x - 2)^2 + (y - 3)^2 + \lambda(x + y - 5) = 0 $$

For different values of $\lambda$, we get different circles in the family.

Example 2: Specific Circle in the Family

Find the equation of a specific circle in the family that passes through the point $(4, 1)$.

Solution:

Using the family equation from Example 1:

$$ (x - 2)^2 + (y - 3)^2 + \lambda(x + y - 5) = 0 $$

Substitute $(4, 1)$ into the equation:

$$ (4 - 2)^2 + (1 - 3)^2 + \lambda(4 + 1 - 5) = 0 $$

Solve for $\lambda$:

$$ 4 + 4 + \lambda(0) = 0 $$

Since the term with $\lambda$ is zero, any value of $\lambda$ will satisfy the equation. This means that the circle with center $(2, 3)$ and radius $2$ already passes through the point $(4, 1)$.

Understanding the family of circles passing through the points of intersection of a line and a circle is crucial for solving various geometric problems. It is also an interesting example of how a single parameter can generate an infinite set of solutions in geometry.