Touching Circles

What Are Touching Circles?

Two circles are said to be touching if they intersect at exactly one point. This point is referred to as the point of contact. Touching circles can either:

Touch Externally: The circles share one tangent at the point of contact, and their centres are on opposite sides of the tangent.
Touch Internally: The smaller circle lies inside the larger circle, and they share a tangent at the point of contact.

Key Properties of Touching Circles

Collinearity of Centres: The centres of the two circles and the point of contact are always collinear.
Tangency Condition:

For external tangency: The distance between the centres of the circles equals the sum of their radii.

|C_1C_2| = r_1 + r_2

For internal tangency: The distance between the centres equals the difference of their radii.

|C_1C_2| = |r_1 - r_2|

Worked Examples

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Example: Verify Touching Circles

Problem: Two circles have equations:

(x - 2)^2 + (y - 3)^2 = 16 \quad \text{and} \quad (x - 6)^2 + (y - 3)^2 = 9

Determine if the circles are touching.

Solution:

Step 1: Identify the centres and radii:

Circle 1: Centre (2, 3), Radius 4
Circle 2: Centre (6, 3), Radius 3

Step 2: Calculate the distance between the centres:

|C_1C_2| = \sqrt{(6 - 2)^2 + (3 - 3)^2} = \sqrt{4^2} = 4

Step 3: Check tangency condition:

|C_1C_2| = r_1 + r_2 = 4 + 3 = 7

The circles are not touching as:

|C_1C_2| \neq r_1 + r_2

Summary

Touching Circles: Circles that intersect at exactly one point.
Conditions for Tangency:
- External Touching: $|C_1C_2| = r_1 + r_2$
- Internal Touching: $|C_1C_2| = |r_1 - r_2|$
Collinearity: The centres of the circles and the point of contact lie on a straight line.
Apply these principles to solve problems involving touching circles and their tangents.

Touching Circles Simplified Revision Notes for Leaving Cert Mathematics