If `|z-1|=1`, then

Options

A. `arg((z-1-i)//z)` can be equal to `-pi//4`
B. `(z-2)//z` is purely imaaginary number
C. `(z-2)//z` is purely real number
D. if `arg(z) = theta,` where `z ne 0` and `theta` is acute, then `1-2//z=itan theta`

Explanation

To solve the problem where \( |z - 1| = 1 \), we will interpret this geometrically and algebraically.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( |z - 1| = 1 \) represents the set of all complex numbers \( z \) that are at a distance of 1 from the point \( 1 \) on the complex plane.

2. **Geometric Interpretation**:
The point \( 1 \) in the complex plane corresponds to the point \( (1, 0) \). The equation describes a circle centered at \( (1, 0) \) with a radius of \( 1 \).

3. **Equation of the Circle**:
The general equation for a circle in the complex plane is given by:
\[ |z - z_0| = r \]
where \( z_0 \) is the center and \( r \) is the radius. Here, \( z_0 = 1 \) and \( r = 1 \).

4. **Converting to Cartesian Coordinates**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then:
\[ |(x + iy) - 1| = 1 \implies |(x - 1) + iy| = 1 \]
This can be expressed as:
\[ \sqrt{(x - 1)^2 + y^2} = 1 \]

5. **Squaring Both Sides**:
Squaring both sides gives:
\[ (x - 1)^2 + y^2 = 1 \]

6. **Expanding the Equation**:
Expanding this, we get:
\[ (x^2 - 2x + 1) + y^2 = 1 \]
Simplifying, we have:
\[ x^2 + y^2 - 2x + 1 - 1 = 0 \implies x^2 + y^2 - 2x = 0 \]

7. **Rearranging**:
Rearranging gives:
\[ x^2 - 2x + y^2 = 0 \]

8. **Completing the Square**:
Completing the square for \( x \):
\[ (x - 1)^2 + y^2 = 1 \]
This confirms that the equation describes a circle centered at \( (1, 0) \) with a radius of \( 1 \).

### Conclusion:
The solution shows that the set of complex numbers \( z \) satisfying \( |z - 1| = 1 \) forms a circle in the complex plane.