If for complex numbers `z_1` and `z_2` and `|1-bar(z_1)z_2|^2-|z_1-z_2|^2=k(1-|z_1|^2)(1-|z_2|^2)` then `k` is equal to:

Options

A. 1
B. `-1`
C. 2
D. 4

Answer

To solve the given problem, we need to analyze the expression step by step. The expression provided is:

\[ |1 - \overline{z_1} z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |z_2|^2) \]

### Step 1: Expand the left-hand side

First, we will expand both terms on the left-hand side.

1. **Calculate \(|1 - \overline{z_1} z_2|^2\)**:
\[ |1 - \overline{z_1} z_2|^2 = (1 - \overline{z_1} z_2)(1 - z_1 \overline{z_2}) = 1 - \overline{z_1} z_2 - z_1 \overline{z_2} + |z_1|^2 |z_2|^2 \]

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

### Step 2: Substitute back into the equation

Now, substituting these expansions back into the original equation:

\[ \left(1 - \overline{z_1} z_2 - z_1 \overline{z_2} + |z_1|^2 |z_2|^2\right) - \left(|z_1|^2 - z_1 \overline{z_2} - z_2 \overline{z_1} + |z_2|^2\right) \]

### Step 3: Simplify the expression

Now, simplify the left-hand side:

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

Combining like terms:

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

### Step 4: Rearranging

Rearranging gives us:

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

### Step 5: Equate to the right-hand side

Now, we need to equate this to the right-hand side:

\[ 1 - |z_1|^2 - |z_2|^2 + |z_1|^2 |z_2|^2 = k(1 - |z_1|^2)(1 - |z_2|^2) \]

### Step 6: Factor the right-hand side

The right-hand side can be expanded as:

\[ k(1 - |z_1|^2)(1 - |z_2|^2) = k(1 - |z_1|^2 - |z_2|^2 + |z_1|^2 |z_2|^2) \]

### Step 7: Compare coefficients

By comparing coefficients from both sides, we can see that:

- The coefficient of \(1\) is \(1\) on the left and \(k\) on the right.
- The coefficient of \(-|z_1|^2\) and \(-|z_2|^2\) is also \(k\).
- The coefficient of \(|z_1|^2 |z_2|^2\) is also \(k\).

This leads us to conclude that:

\[ k = 1 \]

### Final Answer

Thus, the value of \(k\) is:

\[ \boxed{1} \]