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The problem is about a circle \( (C) \) with center \( C \) and radius 3 cm, and a straight line \( (d) \) such that the distance from \( C \) to \( (d) \) is \( l \). You are asked to determine the set of common points \( E \) (i.e., the intersection points between the circle and the line) as a function of \( l \).

Let's break it down:

### 1. Distance \( l \) and Possible Intersections:
- The distance \( l \) refers to how far the line \( (d) \) is from the center of the circle. The key condition here is whether \( l \) is greater than, equal to, or less than the radius of the circle.

- The radius of the circle is 3 cm, so the different cases for \( l \) will be as follows:

### Cases for \( l \):
1. **Case 1: \( l > 3 \)**  
   If the distance from the center of the circle to the line is greater than the radius, the line will not intersect the circle at any point.  
   - Therefore, the set \( E \) will be empty, i.e., \( E = \emptyset \).

2. **Case 2: \( l = 3 \)**  
   If the distance from the center of the circle to the line is exactly equal to the radius, the line will touch the circle at exactly one point.  
   - In this case, the line is tangent to the circle, and the set \( E \) will contain exactly one point, i.e., \( |E| = 1 \).

3. **Case 3: \( l < 3 \)**  
   If the distance from the center of the circle to the line is less than the radius, the line will intersect the circle at two points.  
   - Therefore, the set \( E \) will contain two points, i.e., \( |E| = 2 \).

### Summary:
- If \( l > 3 \), \( E = \emptyset \).
- If \( l = 3 \), \( |E| = 1 \) (the line is tangent).
- If \( l < 3 \), \( |E| = 2 \) (the line intersects at two points).

Would you like more details about any of these cases or how they are derived?

### Relative questions:
1. What happens if the line moves closer to the center of the circle?
2. Can a line intersect a circle at more than two points?
3. How do we compute the distance between a point and a line?
4. What is the formula for the equation of a circle in a plane?
5. How do you find the points of intersection between a circle and a line?

### Tip:
When solving geometry problems involving distances and intersections, sketching the circle and the line on a coordinate plane can help visualize the problem better.

(C) is a circle of center C and radius 3 cm. (d) is a straight line of the plane such that the distance from C to (d) is of given length l. Let E be the set of common points between (C) and (d). Determine E according to the values of l.

Explore the geometric problem of determining the set of common points between a circle with a radius of 3 cm and a line, based on the distance l from the center of the circle to the line. The solution includes key geometric concepts such as line-circle intersections, tangent properties, and distance calculations.

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