Find the value of 'k' if for the complex numbers `z_(1) and z_(2)`.
`|1-bar(z_(1))z_(2)|^(2)-|z_(1)-z_(2)|^(2)=k(1-|z_(1)|^(2))(1-|z_(2)|^(2))`.

Answer

To find the value of \( k \) in the equation

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

we will simplify both sides step by step.

### Step 1: Expand the left-hand side
We start with the left-hand side:

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

Using the property \( |z|^2 = z \overline{z} \), we can rewrite this as:

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

### Step 2: Simplify \( |1 - \overline{z_1} z_2|^2 \)
Calculating \( |1 - \overline{z_1} z_2|^2 \):

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

### Step 3: Simplify \( |z_1 - z_2|^2 \)
Now, calculate \( |z_1 - z_2|^2 \):

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

### Step 4: Combine the results
Now substitute these results back into the left-hand side:

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

This simplifies to:

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

### Step 5: Right-hand side
Now, we simplify the right-hand side:

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

Expanding this gives:

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

### Step 6: Set the two sides equal
Now we set the left-hand side equal to the right-hand side:

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

### Step 7: Solve for \( k \)
To find \( k \), we can compare coefficients. Since both sides are equal for all \( z_1 \) and \( z_2 \), we can deduce that:

\[ k = 1. \]

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

\[ \boxed{1}. \]