If `z_1` and `z_2` are two complex numbers such that `(z_1-2z_2)/(2-z_1bar(z_2))` is unimodular whereas `z_1` is not unimodular then `|z_1|`=

Options

A. Straight line parallel to x-axis
B. sraight line parallel to y-axis
C. circle of radius 2
D. circle of radius `sqrt(2)`

Explanation

To solve the problem step by step, we start by analyzing the given condition that the expression \((z_1 - 2z_2) / (2 - z_1 \overline{z_2})\) is unimodular. A complex number is unimodular if its modulus is equal to 1.

### Step 1: Set up the equation
Given:
\[ \frac{z_1 - 2z_2}{2 - z_1 \overline{z_2}} \text{ is unimodular} \]
This means:
\[ \left| \frac{z_1 - 2z_2}{2 - z_1 \overline{z_2}} \right| = 1 \]

### Step 2: Apply the property of moduli
Using the property of moduli, we can rewrite the equation:
\[ |z_1 - 2z_2| = |2 - z_1 \overline{z_2}| \]

### Step 3: Square both sides
Squaring both sides gives:
\[ |z_1 - 2z_2|^2 = |2 - z_1 \overline{z_2}|^2 \]

### Step 4: Expand both sides
Expanding both sides:
\[ (z_1 - 2z_2)(\overline{z_1} - 2\overline{z_2}) = (2 - z_1 \overline{z_2})(2 - \overline{z_1} z_2) \]

### Step 5: Simplify the left side
The left side simplifies to:
\[ |z_1|^2 - 2z_1 \cdot 2\overline{z_2} - 2\overline{z_1} \cdot z_2 + 4|z_2|^2 \]
This can be written as:
\[ |z_1|^2 + 4|z_2|^2 - 2(z_1 \cdot \overline{z_2} + \overline{z_1} \cdot z_2) \]

### Step 6: Simplify the right side
The right side simplifies to:
\[ 4 - 2(z_1 \overline{z_2} + \overline{z_1} z_2) + |z_1|^2 |z_2|^2 \]

### Step 7: Set the two sides equal
Setting both sides equal gives:
\[ |z_1|^2 + 4|z_2|^2 - 2(z_1 \overline{z_2} + \overline{z_1} z_2) = 4 - 2(z_1 \overline{z_2} + \overline{z_1} z_2) + |z_1|^2 |z_2|^2 \]

### Step 8: Rearrange the equation
Rearranging the equation leads to:
\[ |z_1|^2 + 4|z_2|^2 - |z_1|^2 |z_2|^2 = 4 \]

### Step 9: Factor out common terms
Factoring out gives:
\[ |z_1|^2(1 - |z_2|^2) + 4|z_2|^2 = 4 \]

### Step 10: Solve for \(|z_1|\)
Since \(z_2\) is not unimodular, \(|z_2| \neq 1\). We can rearrange this equation to find \(|z_1|\):
\[ |z_1|^2(1 - |z_2|^2) = 4 - 4|z_2|^2 \]
Thus,
\[ |z_1|^2 = \frac{4(1 - |z_2|^2)}{1 - |z_2|^2} \]
This leads to:
\[ |z_1|^2 = 4 \]
Taking the square root gives:
\[ |z_1| = 2 \]

### Final Answer
\[ |z_1| = 2 \]