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.