If ` z_(1) and z_(2)` are two complex number such that `| (z_(1) - z_(2))/(1 - z_(1) z_(2))| = 1 ` then

Options

A. `|z_(1) | = 1 `
B. ` | z_(2) | = 1 `
C. ` z_(1) = e^(i theta) , theta in R `
D. ` z_(2) = e^(i theta), theta in R`

Answer

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

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

### Step 1: Cross-multiply the modulus equation

From the equation, we can cross-multiply to eliminate the modulus:

\[ |z_1 - z_2| = |1 - z_1 z_2| \]

**Hint:** Remember that if \( |a| = |b| \), then \( a \) and \( b \) can be related through their magnitudes.

### Step 2: Square both sides

Next, we square both sides to remove the modulus:

\[ |z_1 - z_2|^2 = |1 - z_1 z_2|^2 \]

**Hint:** Recall that for any complex number \( z \), \( |z|^2 = z \cdot \overline{z} \).

### Step 3: Expand both sides

Now we expand both sides using the property of modulus:

\[ (z_1 - z_2)(\overline{z_1} - \overline{z_2}) = (1 - z_1 z_2)(1 - \overline{z_1} \overline{z_2}) \]

### Step 4: Calculate the left-hand side

The left-hand side becomes:

\[ (z_1 - z_2)(\overline{z_1} - \overline{z_2}) = |z_1|^2 - z_1 \overline{z_2} - z_2 \overline{z_1} + |z_2|^2 \]

### Step 5: Calculate the right-hand side

The right-hand side expands to:

\[ 1 - z_1 z_2 - \overline{z_1} \overline{z_2} + z_1 z_2 \overline{z_1} \overline{z_2} \]

### Step 6: Set the two sides equal

Now we set the two sides equal:

\[ |z_1|^2 - z_1 \overline{z_2} - z_2 \overline{z_1} + |z_2|^2 = 1 - z_1 z_2 - \overline{z_1} \overline{z_2} + z_1 z_2 \overline{z_1} \overline{z_2} \]

### Step 7: Rearranging terms

Rearranging the equation gives us:

\[ |z_1|^2 + |z_2|^2 - 1 = z_1 z_2 + \overline{z_1} \overline{z_2} + z_1 \overline{z_2} + z_2 \overline{z_1} \]

### Step 8: Analyze the implications

Since \( |z_1|^2 \) and \( |z_2|^2 \) are both non-negative, we can conclude that both must equal 1. Therefore:

\[ |z_1| = 1 \quad \text{and} \quad |z_2| = 1 \]

### Step 9: Conclusion

Thus, we conclude that both \( z_1 \) and \( z_2 \) lie on the unit circle in the complex plane. Therefore, we can express them in the form:

\[ z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2} \quad \text{for some real numbers } \theta_1 \text{ and } \theta_2. \]

### Final Result

The final result is that both complex numbers \( z_1 \) and \( z_2 \) have a modulus of 1.

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