Family of Circles - Always Touching a Straight Line at a Given Point

The concept of the family of circles that always touch a straight line at a given point is an important topic in coordinate geometry, particularly in the study of circles. This family of circles is defined by the common property that all circles in the family are tangent to a given line at a specific point.

Understanding the Family of Circles

A circle is defined as the set of all points in a plane that are at a fixed distance (radius) from a fixed point (center). When we talk about a family of circles, we refer to a set of circles that share a common geometric property.

The General Equation of a Circle

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

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

Equation of a Straight Line

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

[ Ax + By + C = 0 ]

where (A), (B), and (C) are constants.

Point of Tangency

When a circle touches a straight line at exactly one point, that point is called the point of tangency. At this point, the radius of the circle is perpendicular to the tangent line.

Family of Circles Tangent to a Line at a Given Point

To find the family of circles that are tangent to a given line at a point (P(x_1, y_1)), we use the condition that the radius of the circle at the point of tangency is perpendicular to the tangent line.

Condition for Tangency

For a circle with center ((h, k)) to be tangent to the line (Ax + By + C = 0) at point (P(x_1, y_1)), the distance from the center of the circle to the line must be equal to the radius of the circle. This condition is given by:

[ \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} = r ]

Equation of the Family of Circles

The equation of the family of circles tangent to the line (Ax + By + C = 0) at point (P(x_1, y_1)) can be derived by combining the general equation of a circle with the condition for tangency. The resulting equation is:

[ (x - x_1)^2 + (y - y_1)^2 - \lambda(Ax + By + C) = 0 ]

where (\lambda) is a parameter that varies for different circles in the family.

Table of Differences and Important Points

Feature	General Circle	Family of Circles Tangent to a Line
Equation	((x - h)^2 + (y - k)^2 = r^2)	((x - x_1)^2 + (y - y_1)^2 - \lambda(Ax + By + C) = 0)
Center	Fixed at ((h, k))	Varies depending on (\lambda)
Radius	Fixed at (r)	Varies depending on (\lambda)
Tangency	Not necessarily tangent to a line	Always tangent to the line (Ax + By + C = 0) at (P(x_1, y_1))
Parameter	None	(\lambda) determines members of the family

Examples

Example 1: Find the family of circles tangent to the line (3x + 4y - 11 = 0) at the point ((2, 1)).

Using the formula for the family of circles tangent to a line at a given point, we have:

[ (x - 2)^2 + (y - 1)^2 - \lambda(3x + 4y - 11) = 0 ]

This equation represents the family of circles tangent to the line (3x + 4y - 11 = 0) at the point ((2, 1)) for different values of (\lambda).

Example 2: Determine a specific circle from the family that has a radius of 5.

Given the family of circles from Example 1, we need to find the value of (\lambda) such that the radius is 5. Using the condition for tangency:

[ r = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} ]

Substituting (r = 5), (A = 3), (B = 4), and (C = -11), and the coordinates of the center ((h, k)) as ((2, 1)), we get:

[ 5 = \frac{|3 \cdot 2 + 4 \cdot 1 - 11|}{\sqrt{3^2 + 4^2}} ]

Solving for (\lambda), we find the specific circle in the family with a radius of 5.

By understanding the family of circles tangent to a line at a given point, students can solve various problems related to circle geometry and tangency conditions. This topic is essential for examinations that include coordinate geometry and circle theorems.