Illustrate in the complex plane the following set of points and explain your answer
`|Z|=3`

Answer

To illustrate the set of points defined by the equation \( |Z| = 3 \) in the complex plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Complex Number**:
Let \( Z \) be a complex number represented as \( Z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

2. **Using the Modulus Definition**:
The modulus of a complex number \( Z \) is given by:
\[ |Z| = \sqrt{x^2 + y^2} \]
According to the problem, we have:
\[ |Z| = 3 \]

3. **Setting Up the Equation**:
From the modulus definition, we can set up the equation:
\[ \sqrt{x^2 + y^2} = 3 \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ x^2 + y^2 = 3^2 \]
This simplifies to:
\[ x^2 + y^2 = 9 \]

5. **Identifying the Geometric Shape**:
The equation \( x^2 + y^2 = 9 \) represents a circle in the Cartesian coordinate system. The standard form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) \) is the center and \( r \) is the radius. Here, the center is at \( (0, 0) \) and the radius \( r \) is 3.

6. **Graphing the Circle**:
To graph this circle in the complex plane (also known as the Argand plane), we plot the center at the origin \( (0, 0) \) and draw a circle with a radius of 3 units. The points on this circle represent all the complex numbers \( Z \) such that \( |Z| = 3 \).

### Conclusion:
The set of points defined by \( |Z| = 3 \) is a circle centered at the origin (0, 0) with a radius of 3 units in the complex plane.