Condition for Two Circles Touching and Intersecting Each Other

Two circles can either be separate, touch each other, or intersect. The relationship between two circles is determined by the distance between their centers and the sum or difference of their radii.

Definitions

• Center: The fixed point from which all points on the circle are equidistant.
• Radius: The distance from the center of the circle to any point on the circle.
• Externally Tangent: Two circles that touch each other at exactly one point and lie outside each other.
• Internally Tangent: Two circles that touch each other at exactly one point with one circle inside the other.
• Intersecting Circles: Two circles that meet at two distinct points.

Conditions

Touching Circles

Two circles touch each other if the distance between their centers is equal to the sum (for externally tangent) or the absolute difference (for internally tangent) of their radii.

Externally Tangent Circles

For two circles with centers (C_1) and (C_2) and radii (r_1) and (r_2), the condition for them to be externally tangent is:

[ d = r_1 + r_2 ]

where (d) is the distance between the centers (C_1) and (C_2).

Internally Tangent Circles

For internally tangent circles, the condition is:

[ d = |r_1 - r_2| ]

Intersecting Circles

Two circles intersect if the distance between their centers is less than the sum of their radii and greater than the absolute difference of their radii.

[ |r_1 - r_2| < d < r_1 + r_2 ]

Table of Differences and Important Points

Condition	Description	Formula	Example
Externally Tangent	Touch at one point, no common interior points	(d = r_1 + r_2)	Two circles with radii 3 and 5 units, centers 8 units apart
Internally Tangent	Touch at one point, one circle inside the other	(d =	r_1 - r_2
Intersecting	Meet at two points, have common interior points	(	r_1 - r_2

Examples

Example 1: Externally Tangent Circles

Consider two circles, one with radius (r_1 = 3) units and the other with radius (r_2 = 5) units. For these circles to be externally tangent, the distance between their centers must be:

[ d = r_1 + r_2 = 3 + 5 = 8 \text{ units} ]

Example 2: Internally Tangent Circles

Using the same radii as above, for the circles to be internally tangent, the distance between their centers must be:

[ d = |r_1 - r_2| = |3 - 5| = 2 \text{ units} ]

Example 3: Intersecting Circles

For the circles to intersect, the distance between their centers must satisfy:

[ |r_1 - r_2| < d < r_1 + r_2 ] [ |3 - 5| < d < 3 + 5 ] [ 2 < d < 8 ]

So if the distance between the centers is, for example, 6 units, the circles will intersect.

Conclusion

The relationship between two circles—whether they are separate, touching, or intersecting—depends on the distance between their centers relative to their radii. By understanding and applying the conditions and formulas provided, one can determine the exact nature of the interaction between any two circles.