If z complex number satisfying `|z-1| = 1`, then which of the following is correct

Options

A. `"arg"(z-1)=2"arg"(z)`
B. `2arg(z) =2/3"arg"(z^(2)-z)`
C. `"arg"(z-1)=2"arg"(z+1)`
D. `"arg"z=2"arg"(z+1)`

Explanation

To solve the problem, we need to analyze the given condition for the complex number \( z \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that \( |z - 1| = 1 \). This means that the distance between the complex number \( z \) and the point \( 1 \) (which corresponds to the complex number \( 1 + 0i \)) in the complex plane is equal to \( 1 \).

**Hint**: Recall that the modulus of a complex number represents the distance from the origin in the complex plane.

2. **Identifying the Geometric Representation**:
The equation \( |z - 1| = 1 \) describes a circle in the complex plane. The center of this circle is at the point \( 1 + 0i \) (or simply \( 1 \) on the real axis), and the radius of the circle is \( 1 \).

**Hint**: A circle in the complex plane can be represented as \( |z - z_0| = r \), where \( z_0 \) is the center and \( r \) is the radius.

3. **Finding the Center and Radius**:
- Center: \( (1, 0) \)
- Radius: \( 1 \)

This means the circle passes through the points \( 0 \) (the origin) and \( 2 \) (the point on the real axis at \( 2 \)).

**Hint**: Sketching the circle can help visualize the problem better.

4. **Analyzing Points on the Circle**:
Let \( z \) be any point on this circle. We can express \( z \) in terms of its argument and modulus. If we let \( z = re^{i\theta} \), we can find the argument of \( z \).

**Hint**: Remember that the argument of a complex number is the angle it makes with the positive real axis.

5. **Relating the Argument of \( z \) and \( z - 1 \)**:
The argument of \( z - 1 \) can be expressed in terms of the angle \( \theta \) formed by the line connecting the center of the circle to the point \( z \). If we denote the angle \( XOP \) as \( \theta \), then the angle \( XCP \) (where \( C \) is the center) can be related to the argument of \( z \).

**Hint**: Use properties of angles in circles to relate the angles.

6. **Establishing the Relationship**:
From the geometry, we can conclude that:
\[ \text{arg}(z - 1) = 2 \cdot \text{arg}(z) \]
This means the argument of \( z - 1 \) is twice the argument of \( z \).

**Hint**: This relationship is crucial for determining the correct option from the given choices.

7. **Matching with Options**:
Finally, we need to compare this result with the provided options to determine which one is correct. Based on our analysis, we find that:
\[ \text{arg}(z - 1) = 2 \cdot \text{arg}(z) \]
is indeed one of the options.

**Hint**: Always double-check your options against the derived relationships.

### Conclusion:
The correct option is that \( \text{arg}(z - 1) = 2 \cdot \text{arg}(z) \).