Position of Two Circles

Relationship Between Circles

Have you ever noticed how two soap bubbles interact? Sometimes they intersect, sometimes they just touch briefly, or they might even avoid each other completely. Well, the mathematical concept of position of two circles is really similar to this phenomenon!

In analytic geometry, we can determine with certainty how two circles relate to each other: whether they intersect, are tangent, or are completely separate. What's interesting is that all of this can be predicted just by knowing the center and radius of each circle.

This concept is super useful in real life. For example, to design gears that must be tangent perfectly, calculate the coverage area of two radio antennas, or even plan a garden with round ponds that are interconnected.

Intersecting Circles

Two circles are said to be intersecting if they meet at two different points. Just imagine two rings that "penetrate" each other.

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

Here's the logic:

• Upper bound: If center distance = $r_1 + r_2$, both circles only touch externally
• Lower bound: If center distance = $|r_1 - r_2|$, the small circle touches the large one internally
• Intersection area: Between these two bounds, circles definitely intersect at two points

Tangent Circles

Tangent means two circles only meet at one point. Like two wheels that touch at exactly one point to transfer motion.

There are two types of tangency:

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

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

Here's an example of internally tangent circles:

Separate Circles

This condition occurs when the two circles don't touch at all. Like two islands separated by ocean, there's no physical connection between them.

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

In this situation, there's no point that belongs to both circles simultaneously. They are completely separate in the coordinate plane.

Concentric and Coincident Circles

Concentric circles are two circles that have the same center but different radii. Imagine an archery target with circles that have the same center.

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

Coincident circles are a special condition where both circles are completely identical. They have the same center and radius, so they look like just one circle.

The coincident condition occurs when:

How to Determine Position

To determine the position of two circles practically, we need to calculate the distance between centers and compare it with the 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:

After getting the value $d$, we can determine the position based on the following conditions:

• Separate: $d > r_1 + r_2$ (circles far apart)
• Externally tangent: $d = r_1 + r_2$ (touching externally)
• Intersecting: $|r_1 - r_2| < d < r_1 + r_2$ (intersecting at two points)
• Internally tangent: $d = |r_1 - r_2|$ (touching internally)
• Non-intersecting: $d < |r_1 - r_2|$ (one circle inside the other)
• Concentric: $d = 0$ and $r_1 \ne r_2$ (same center, different radii)
• Coincident: $d = 0$ and $r_1 = r_2$ (identical circles)

Application Example

Determine the position of two 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.

To ensure the answer is correct, we can check the condition $|r_1 - r_2| < d < r_1 + r_2$:

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