Draw the circles of different radii. How many points, these circles can have in common ? What is the maximum number of common points ?

Answer

To solve the question step by step, we will analyze the situation of two circles with different radii and determine how many points they can have in common.

### Step 1: Draw Two Circles
- Draw two circles on a plane. Let's denote the first circle as Circle A with radius R1 and the second circle as Circle B with radius R2, where R1 < R2.

### Step 2: Analyze the Position of the Circles
- We need to consider the relative positions of the two circles to determine how many points they can have in common. There are three possible cases:

### Step 3: Case 1 - No Intersection
- If the distance between the centers of Circle A and Circle B is greater than the sum of their radii (i.e., distance > R1 + R2), the circles do not intersect.
- **Common Points:** 0 points.

### Step 4: Case 2 - Tangential Intersection
- If the distance between the centers of the circles is exactly equal to the sum of their radii (i.e., distance = R1 + R2), the circles touch at exactly one point. This is known as external tangency.
- **Common Points:** 1 point.

### Step 5: Case 3 - Intersecting Circles
- If the distance between the centers of the circles is less than the sum of their radii but greater than the absolute difference of their radii (i.e., |R1 - R2| < distance < R1 + R2), the circles intersect at two points.
- **Common Points:** 2 points.

### Step 6: Maximum Number of Common Points
- From the analysis above, we can summarize the maximum number of common points:
- Case 1: 0 points (no intersection)
- Case 2: 1 point (tangential intersection)
- Case 3: 2 points (two points of intersection)
- Therefore, the maximum number of common points that two circles can have is **2 points**.

### Final Answer
The maximum number of common points between two circles of different radii is **2 points**.

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