If `z_(0),z_(1)` represent points P and Q on the circle `|z-1|=1` taken in anticlockwise sense such that the line segment PQ subtends a right angle at the center of the circle, then `z_(1)=`

Options

A. `1+i(z_(0)-1)`
B. `iz_(0)`
C. `1-i(z_(0)-1)`
D. `i(z_(0)-1)`

Explanation

To solve the problem, we need to find the point \( z_1 \) on the circle defined by \( |z - 1| = 1 \) such that the line segment \( PQ \) (where \( P \) is represented by \( z_0 \) and \( Q \) by \( z_1 \)) subtends a right angle at the center of the circle.

### Step-by-step Solution:

1. **Understanding the Circle**:
The equation \( |z - 1| = 1 \) represents a circle centered at \( (1, 0) \) with a radius of 1. In the complex plane, this can be represented as:
\[ z = 1 + e^{i\theta} \]
where \( \theta \) varies from \( 0 \) to \( 2\pi \).

2. **Identifying Points**:
Let \( z_0 \) be the point \( P \) on the circle, which can be expressed as:
\[ z_0 = 1 + e^{i\alpha} \]
for some angle \( \alpha \). The point \( z_1 \) (point \( Q \)) will also lie on the same circle:
\[ z_1 = 1 + e^{i\beta} \]
for some angle \( \beta \).

3. **Condition of Right Angle**:
The line segment \( PQ \) subtends a right angle at the center of the circle. This means that the angle \( \angle PCQ \) is \( \frac{\pi}{2} \). In terms of complex numbers, this implies:
\[ (z_1 - 1) \cdot \overline{(z_0 - 1)} + (z_0 - 1) \cdot \overline{(z_1 - 1)} = 0 \]
This can be simplified to:
\[ (e^{i\beta}) \cdot (e^{-i\alpha}) + (e^{i\alpha}) \cdot (e^{-i\beta}) = 0 \]
which leads to:
\[ e^{i(\beta - \alpha)} + e^{i(\alpha - \beta)} = 0 \]
This implies:
\[ e^{i(\beta - \alpha)} = -1 \]
Therefore, we have:
\[ \beta - \alpha = \pi \quad \text{(mod } 2\pi\text{)} \]

4. **Finding \( z_1 \)**:
From the above condition, we can express \( \beta \) in terms of \( \alpha \):
\[ \beta = \alpha + \pi \]
Substituting this back into the expression for \( z_1 \):
\[ z_1 = 1 + e^{i(\alpha + \pi)} = 1 - e^{i\alpha} \]

5. **Final Expression**:
Since \( e^{i\alpha} = z_0 - 1 \), we can rewrite \( z_1 \) as:
\[ z_1 = 1 - (z_0 - 1) = 2 - z_0 \]

Thus, the final answer is:
\[ z_1 = 2 - z_0 \]