https://file.aiquickdraw.com/mathbot/file/172594542828750wmctvv.jpg

The text in the image is in Persian and translates to:

"Two points A and B are located 8 cm apart. How many points exist in the plane that are 4 cm from point A and 2 cm from point B?"

### Solution:

This problem is related to the geometric concept of **locus**. Specifically, it is asking for points that satisfy two distance conditions from two fixed points.

- Let \( A \) and \( B \) be two points separated by a distance of 8 cm.
- We are looking for the points that are 4 cm away from \( A \) and 2 cm away from \( B \).

These points lie on **circles** centered at \( A \) and \( B \):
- A circle with radius 4 cm centered at \( A \).
- A circle with radius 2 cm centered at \( B \).

The number of points that satisfy both conditions is the number of **intersections** of these two circles. 

#### Approach:
To solve this, we need to determine how many times the two circles intersect:
1. **Distance between centers**: The distance between \( A \) and \( B \) is 8 cm.
2. **Sum of radii**: The sum of the radii is \( 4 + 2 = 6 \) cm.
3. **Difference of radii**: The difference of the radii is \( 4 - 2 = 2 \) cm.

### Intersection conditions:
- If the distance between the centers (8 cm) is greater than the sum of the radii (6 cm), the circles do **not intersect**.
- If the distance between the centers (8 cm) is exactly equal to the sum of the radii (6 cm), the circles touch at **one point** (externally tangent).
- If the distance between the centers (8 cm) is less than the sum of the radii (6 cm) but greater than the difference of the radii (2 cm), the circles intersect at **two points**.

Given that the distance between \( A \) and \( B \) is **greater than the sum of the radii**, the circles do not intersect.

### Conclusion:
No points satisfy both distance conditions. Therefore, **there are no such points** in the plane.

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Would you like further details on this? Here are some additional related questions:

1. What is the general condition for two circles to intersect in a plane?
2. Can we have such a configuration where the circles would intersect? How would the distances need to change?
3. How is the concept of the locus used in other geometrical problems?
4. What happens if the difference between the centers of the two circles equals the difference in their radii?
5. How can we solve similar problems using algebraic methods?

**Tip**: In geometry, circle intersection problems are often solved by comparing the sum and difference of radii with the distance between the centers.

Two points A and B are located 8 cm apart. How many points exist in the plane that are 4 cm from point A and 2 cm from point B?

In this geometry problem, we explore how to find points that are 4 cm from point A and 2 cm from point B, where the points A and B are 8 cm apart. The solution involves understanding the conditions for circle intersection and the concept of locus. Suitable for students in Grades 10-12.

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