Condition of Tangency

The condition of tangency refers to the specific criteria that must be met for a line to be tangent to a circle. A tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. Understanding the condition of tangency is crucial for solving problems related to circles and tangents in geometry.

Basic Properties of a Tangent

Before we delve into the condition of tangency, let's review some basic properties of a tangent to a circle:

• A tangent to a circle is perpendicular to the radius at the point of tangency.
• A tangent never crosses or enters the circle; it only touches the circle at one point.
• For a given external point, there can be exactly two tangents to a circle.

Condition of Tangency for a Line and a Circle

The condition of tangency for a line and a circle can be expressed in terms of the distance from the center of the circle to the line. Let's consider a circle with center (O) and radius (r), and a line with the equation (ax + by + c = 0). The distance (d) from the center of the circle ((h, k)) to the line is given by the formula:

[ d = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} ]

For the line to be tangent to the circle, the distance (d) must be equal to the radius (r) of the circle. Therefore, the condition of tangency is:

[ \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} = r ]

This can also be written as:

[ |ah + bk + c|^2 = r^2(a^2 + b^2) ]

Condition of Tangency for Two Circles

When considering two circles, the condition of tangency relates to the distance between their centers and the sum or difference of their radii. Let's consider two circles with centers (O_1) and (O_2), and radii (r_1) and (r_2) respectively. The distance (d) between the centers is:

[ d = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2} ]

For the circles to be externally tangent, the condition is:

[ d = r_1 + r_2 ]

For the circles to be internally tangent, the condition is:

[ d = |r_1 - r_2| ]

Table of Differences and Important Points

Property	Line and Circle Tangency	Two Circles Tangency
Definition	A line is tangent to a circle if it touches the circle at exactly one point.	Two circles are tangent to each other if they touch at exactly one point.
Condition	( \frac{	ah + bk + c
Geometric Relation	The tangent is perpendicular to the radius at the point of tangency.	No such perpendicularity condition exists between the radii of the two circles.
Number of Tangents	From an external point, two tangents can be drawn to a circle.	Two circles can have at most one common tangent at the point of tangency.

Examples

Example 1: Line and Circle Tangency

Consider a circle with center at (O(2, 3)) and radius (r = 5), and a line with the equation (3x + 4y + c = 0). To find the value of (c) for which the line is tangent to the circle, we use the condition of tangency:

[ \frac{|3(2) + 4(3) + c|}{\sqrt{3^2 + 4^2}} = 5 ]

Solving for (c), we get:

[ \frac{|6 + 12 + c|}{5} = 5 ]

[ |18 + c| = 25 ]

[ c = 7 \text{ or } c = -43 ]

Thus, the line (3x + 4y + 7 = 0) or (3x + 4y - 43 = 0) is tangent to the circle.

Example 2: Two Circles Tangency

Consider two circles with centers (O_1(1, 2)) and (O_2(4, 6)), and radii (r_1 = 3) and (r_2 = 2). To determine if they are tangent to each other, we calculate the distance between their centers:

[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} ]

[ d = \sqrt{3^2 + 4^2} ]

[ d = \sqrt{9 + 16} ]

[ d = \sqrt{25} ]

[ d = 5 ]

Since (d = r_1 + r_2 = 3 + 2 = 5), the circles are externally tangent to each other.

Understanding the condition of tangency is essential for solving geometric problems involving circles and tangents. By applying the formulas and conditions outlined above, one can determine whether a line is tangent to a circle or if two circles are tangent to each other.