If `z_1,z_2` are complex numbers such that, `|(z_1-3z_2)/(3-z_1.bar z_2)|=1` and `|z_2|ne 1` then find `|z_1|`

Answer

To solve the problem, we start with the given condition involving the complex numbers \( z_1 \) and \( z_2 \):

\[ \left| \frac{z_1 - 3z_2}{3 - z_1 \overline{z_2}} \right| = 1 \]

### Step 1: Rewrite the modulus equation
Using the property of moduli, we can separate the numerator and denominator:

\[ |z_1 - 3z_2| = |3 - z_1 \overline{z_2}| \]

### Step 2: Square both sides
To eliminate the modulus, we square both sides:

\[ |z_1 - 3z_2|^2 = |3 - z_1 \overline{z_2}|^2 \]

### Step 3: Expand both sides
Using the property that \( |z|^2 = z \overline{z} \), we expand both sides:

\[ (z_1 - 3z_2)(\overline{z_1} - 3\overline{z_2}) = (3 - z_1 \overline{z_2})(3 - \overline{z_1} z_2) \]

### Step 4: Simplify the left-hand side
Expanding the left-hand side:

\[ z_1 \overline{z_1} - 3z_1 \overline{z_2} - 3z_2 \overline{z_1} + 9z_2 \overline{z_2} \]

### Step 5: Simplify the right-hand side
Expanding the right-hand side:

\[ 9 - 3z_1 \overline{z_2} - 3\overline{z_1} z_2 + z_1 \overline{z_1} z_2 \overline{z_2} \]

### Step 6: Set both sides equal
Setting the expanded forms equal gives us:

\[ z_1 \overline{z_1} - 3z_1 \overline{z_2} - 3z_2 \overline{z_1} + 9|z_2|^2 = 9 - 3z_1 \overline{z_2} - 3\overline{z_1} z_2 + |z_1|^2 |z_2|^2 \]

### Step 7: Rearranging terms
Rearranging the terms leads to:

\[ |z_1|^2 - 9|z_2|^2 + 9 - |z_1|^2 |z_2|^2 = 0 \]

### Step 8: Factor the equation
Factoring gives us:

\[ |z_1|^2(1 - |z_2|^2) - 9(1 - |z_2|^2) = 0 \]

### Step 9: Solve for \( |z_1|^2 \)
Factoring out \( (1 - |z_2|^2) \):

\[ (1 - |z_2|^2)(|z_1|^2 - 9) = 0 \]

Since we know \( |z_2| \neq 1 \), it follows that \( 1 - |z_2|^2 \neq 0 \). Therefore, we must have:

\[ |z_1|^2 - 9 = 0 \]

### Step 10: Final result
This leads us to:

\[ |z_1|^2 = 9 \implies |z_1| = 3 \]

Thus, the final answer is:

\[ |z_1| = 3 \]