Condition for Two Circles Touching Each Other

Two circles can touch each other in two different ways: externally and internally. When two circles touch each other externally, they have one common point and lie outside each other. When they touch each other internally, they also have one common point, but one circle lies inside the other.

Important Points and Differences

Aspect	External Tangency	Internal Tangency
Position of Circles	Circles lie outside each other	One circle lies inside the other
Common Tangent	Touch at one point from outside	Touch at one point from inside
Distance Between Centers	Equal to the sum of their radii	Equal to the difference of their radii
Formula	$d = r_1 + r_2$	$d =
Example	Two non-overlapping soap bubbles	A small bubble inside a larger bubble

Formulas

Let's denote the centers of the two circles as $C_1(x_1, y_1)$ and $C_2(x_2, y_2)$, and their radii as $r_1$ and $r_2$ respectively.

External Tangency

For two circles to touch each other externally, the distance between their centers ($d$) must be equal to the sum of their radii.

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = r_1 + r_2 $$

Internal Tangency

For two circles to touch each other internally, the distance between their centers must be equal to the absolute difference of their radii.

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = |r_1 - r_2| $$

Examples

Example 1: External Tangency

Consider two circles with centers $C_1(2, 3)$ and $C_2(7, 8)$ and radii $r_1 = 3$ and $r_2 = 2$ respectively.

To check if they touch each other externally, we calculate the distance between their centers:

$$ d = \sqrt{(7 - 2)^2 + (8 - 3)^2} = \sqrt{5^2 + 5^2} = \sqrt{50} \approx 7.07 $$

Since $r_1 + r_2 = 3 + 2 = 5$, and $d \approx 7.07$, these circles do not touch each other externally.

Example 2: Internal Tangency

Consider two circles with centers $C_1(1, 1)$ and $C_2(4, 5)$ and radii $r_1 = 5$ and $r_2 = 2$ respectively.

To check if they touch each other internally, we calculate the distance between their centers:

$$ d = \sqrt{(4 - 1)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 $$

Since $|r_1 - r_2| = |5 - 2| = 3$, and $d = 5$, these circles do not touch each other internally either.

However, if the radius of the second circle ($r_2$) was $3$ instead of $2$, then the circles would touch internally because $|r_1 - r_2| = |5 - 3| = 2$ and $d = 5$.

Conclusion

The condition for two circles to touch each other is determined by the distance between their centers and the sum or the absolute difference of their radii, depending on whether the tangency is external or internal. Understanding these conditions is crucial for solving problems related to circle geometry, especially in the context of exams where precise knowledge and application of these concepts are required.