Illustrate and explain the set of points z in the Argand diagram, which represents `|z- z_(1)| le 3` where `z_(1)= 3-2i`

Answer

To solve the problem of illustrating and explaining the set of points \( z \) in the Argand diagram that represents \( |z - z_1| \leq 3 \) where \( z_1 = 3 - 2i \), follow these steps:

### Step-by-Step Solution:

1. **Identify the Complex Number**:
We have \( z_1 = 3 - 2i \). This means that the point \( z_1 \) corresponds to the coordinates \( (3, -2) \) in the Argand diagram (where the x-axis represents the real part and the y-axis represents the imaginary part).

2. **Understand the Modulus Condition**:
The expression \( |z - z_1| \leq 3 \) represents the distance from the point \( z \) to the point \( z_1 \). The modulus \( |z - z_1| \) gives the distance between the point \( z \) and the point \( z_1 \).

3. **Interpret the Inequality**:
The inequality \( |z - z_1| \leq 3 \) means that the distance from any point \( z \) to the point \( z_1 \) should be less than or equal to 3. This describes all points that are within (or on the boundary of) a circle centered at \( z_1 \) with a radius of 3.

4. **Draw the Argand Diagram**:
- Draw the Cartesian plane with the real axis (x-axis) and the imaginary axis (y-axis).
- Mark the center of the circle at the point \( (3, -2) \).
- Since the radius is 3, you will draw a circle around the point \( (3, -2) \) with a radius of 3.

5. **Complete the Circle**:
- The circle will include all points that are exactly 3 units away from \( (3, -2) \) and all points inside this circle.
- The equation of the circle can be expressed as \( (x - 3)^2 + (y + 2)^2 = 9 \), where \( (x, y) \) are the coordinates of the points \( z \).

6. **Final Representation**:
- The final representation in the Argand diagram will show a filled circle (including the boundary) centered at \( (3, -2) \) with a radius of 3.