Position of Two Circles

Comparing the Distance Between Centers with the Radii

The circumferences of two circles may cross at two points, touch at one point, or remain separate. Their disks may therefore overlap, meet at one boundary point, or leave a gap.

In analytic geometry, the centers and radii are enough to determine the relationship precisely. The decisive quantity is the distance between the two centers.

The same comparison appears when checking whether circular regions overlap, touch, or leave a gap, for example in coverage maps or geometric designs.

Intersecting Circles

Two circles are intersecting when they have exactly two common points. Their disks overlap, while their circumferences cross at those two points.

For two circles with radii $r_1$ and $r_2$ and distance between centers $d$, the intersection condition occurs when:

The two strict bounds explain the condition:

• Upper bound: At $r_1 + r_2$, the circles are externally tangent
• Lower bound: At $|r_1 - r_2|$, the smaller circle is internally tangent to the larger circle
• Between the bounds: The circumferences cross at two points

Tangent Circles

Two circles are tangent when they have exactly one common point. The tangency may occur outside both circles or inside the larger circle.

For two noncoincident circles, there are two tangency cases:

1. External tangency occurs when $d = r_1 + r_2$. Both circles are separate and touch at one point.

2. Internal tangency occurs when $d = |r_1 - r_2| > 0$. The smaller circle is inside the larger one and they touch at one point.

The next diagram shows internal tangency:

Separate Circles

Two circles are externally separate when neither their circumferences nor their disks meet.

The separate condition occurs when the distance between centers is greater than the sum of both radii:

In this case, no point belongs to both disks.

Concentric and Coincident Circles

Concentric circles have the same center but different radii, like the circular boundaries on an archery target.

For concentric circles, the distance between centers is zero ($d = 0$) but the radii are different ($r_1 \ne r_2$).

Coincident circles have the same center and the same radius. Their circumferences lie exactly on top of each other, so they appear as one circle.

The coincident condition occurs when:

How to Determine Position

To classify two circles, calculate the distance between their centers and compare it with the sum and absolute difference of their radii.

Suppose the first circle is centered at $(x_1, y_1)$ with radius $r_1$, and the second circle is centered at $(x_2, y_2)$ with radius $r_2$.

The distance between centers is calculated using the formula:

Assuming both radii are positive, the following conditions are exhaustive and do not overlap:

• Coincident: $d = 0$ and $r_1 = r_2$ (the same circumference)
• Concentric but distinct: $d = 0$ and $r_1 \ne r_2$ (the same center, different radii)
• Contained without touching and nonconcentric: $0 < d < |r_1-r_2|$ (one circle lies strictly inside the other)
• Internally tangent: $d = |r_1-r_2| > 0$ (one internal point of contact)
• Intersecting: $|r_1-r_2| < d < r_1+r_2$ (two common points)
• Externally tangent: $d = r_1+r_2$ (one external point of contact)
• Externally separate: $d > r_1+r_2$ (the two disks do not meet)

Classifying Two Circles from Their Equations

Classify the circles with equations $x^2 + y^2 = 9$ and $x^2 + y^2 - 6x - 8y = 0$.

Step 1: Identify the center and radius of each circle.

First circle: center $(0, 0)$, radius $r_1 = 3$

For the second circle, we complete the square:

Second circle: center $(3, 4)$, radius $r_2 = 5$

Step 2: Calculate the distance between centers.

Step 3: Compare with position conditions.

Since $2 < 5 < 8$, the two circles are intersecting.

As a check, substitute the three values into $|r_1 - r_2| < d < r_1 + r_2$:

• $|3 - 5| = 2$
• $3 + 5 = 8$
• $2 < 5 < 8$ $\checkmark$ (intersection condition satisfied)