If the two circles `C_(1) and C_(2)` have three points in common, then which of the following is correct ?

Options

A. `C_(1) and C_(2)` are concentric
B. `C_(1) and C_(2)` are the same circle
C. `C_(1) and C_(2)` have different centres
D. None of the above

Answer

To solve the question, we need to analyze the relationship between two circles and the number of points they can have in common.

### Step-by-Step Solution:

1. **Understanding Circle Intersections**:
- Two circles can intersect in different ways. They can touch at one point, intersect at two points, be concentric (having the same center but different radii), or be the same circle.

2. **Identifying Cases**:
- **Case 1**: Two circles touch at one point (1 common point).
- **Case 2**: Two circles intersect at two points (2 common points).
- **Case 3**: One circle is concentric to the other (0 common points).
- **Case 4**: Two circles are the same (infinite common points).

3. **Analyzing the Given Condition**:
- The question states that the two circles \( C_1 \) and \( C_2 \) have three points in common.

4. **Evaluating the Cases**:
- From the cases outlined, we see that none of the cases allows for exactly three points in common.
- The only scenario where there are an infinite number of common points is when the two circles are the same.

5. **Conclusion**:
- Since having three common points is not possible under the defined cases, the only conclusion we can draw is that if two circles have three points in common, they must be the same circle, which means they have infinite points in common.

### Final Answer:
- Therefore, the correct option is that \( C_1 \) and \( C_2 \) are the same circle.