If `z_1` and `z_2` are complex numbers such that, `|1-bar z_1z_2|^(2)-|z_1-z_(2^(2))|^(2)=k(1-|z_1|^(2)) (1-|z_2|^(2))`.Find value of k.

Answer

To solve the problem, we need to analyze the given equation:

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

### Step 1: Expand the Left-Hand Side (LHS)

We start with the left-hand side:

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

Using the property of modulus, we can express this as:

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

Next, we will expand the second term:

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

### Step 2: Combine the Terms

Now, we combine these two results:

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

### Step 3: Simplify the Expression

Now, we simplify the expression:

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

Combining like terms, we have:

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

### Step 4: Factor Out Common Terms

Now, we notice that we can factor out \( (1 - |z_1|^2)(1 - |z_2|^2) \):

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

### Step 5: Compare with the Right-Hand Side (RHS)

Now we have:

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

Comparing this with the right-hand side of the original equation:

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

### Step 6: Determine the Value of k

From the comparison, we see that:

\[ k = 1 \]

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

\[ \boxed{1} \]