Let `z_(1), z_(2)` be two non-zero complex numbers such that `|z_(1) + z_(2)| = |z_(1) - z_(2)|`, then `(z_(1))/(bar(z)_(1)) + (z_(2))/(bar(z)_(2))` equals

Options

A. 0
B. 1
C. -1
D. None of these

Answer

To solve the problem, we start with the given condition for two non-zero complex numbers \( z_1 \) and \( z_2 \):

Given:
\[ |z_1 + z_2| = |z_1 - z_2| \]

We need to find the value of:
\[ \frac{z_1}{\bar{z_1}} + \frac{z_2}{\bar{z_2}} \]

### Step 1: Express \( z_1 \) and \( z_2 \) in terms of their real and imaginary parts
Let:
\[ z_1 = x_1 + i y_1 \quad \text{and} \quad z_2 = x_2 + i y_2 \]

### Step 2: Use the modulus condition
From the condition \( |z_1 + z_2| = |z_1 - z_2| \), we can write:
\[ | (x_1 + x_2) + i (y_1 + y_2) | = | (x_1 - x_2) + i (y_1 - y_2) | \]

### Step 3: Write the modulus in terms of real and imaginary parts
The modulus of a complex number \( a + ib \) is given by:
\[ |a + ib| = \sqrt{a^2 + b^2} \]
Thus, we have:
\[ \sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (x_1 + x_2)^2 + (y_1 + y_2)^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 \]

### Step 5: Expand both sides
Expanding both sides:
\[ x_1^2 + 2x_1x_2 + x_2^2 + y_1^2 + 2y_1y_2 + y_2^2 = x_1^2 - 2x_1x_2 + x_2^2 + y_1^2 - 2y_1y_2 + y_2^2 \]

### Step 6: Simplify the equation
Canceling common terms:
\[ 2x_1x_2 + 2y_1y_2 = -2x_1x_2 - 2y_1y_2 \]
This simplifies to:
\[ 4x_1x_2 + 4y_1y_2 = 0 \]
Thus:
\[ x_1x_2 + y_1y_2 = 0 \]

### Step 7: Substitute into the expression we need to evaluate
Now we need to evaluate:
\[ \frac{z_1}{\bar{z_1}} + \frac{z_2}{\bar{z_2}} \]
We know:
\[ \frac{z_1}{\bar{z_1}} = \frac{x_1 + iy_1}{x_1 - iy_1} = \frac{(x_1 + iy_1)(x_1 + iy_1)}{x_1^2 + y_1^2} = \frac{x_1^2 + y_1^2 + 2ix_1y_1}{x_1^2 + y_1^2} = 1 + \frac{2iy_1}{x_1^2 + y_1^2} \]
Similarly,
\[ \frac{z_2}{\bar{z_2}} = 1 + \frac{2iy_2}{x_2^2 + y_2^2} \]

### Step 8: Combine the results
Thus:
\[ \frac{z_1}{\bar{z_1}} + \frac{z_2}{\bar{z_2}} = 2 + \frac{2i(y_1 + y_2)}{x_1^2 + y_1^2} + \frac{2i(y_2)}{x_2^2 + y_2^2} \]

### Step 9: Use the condition \( x_1x_2 + y_1y_2 = 0 \)
From our earlier result, since \( x_1x_2 + y_1y_2 = 0 \), we can conclude that \( y_1 + y_2 = 0 \).

### Final Result
Thus, substituting \( y_1 + y_2 = 0 \) into our expression gives:
\[ \frac{z_1}{\bar{z_1}} + \frac{z_2}{\bar{z_2}} = 0 \]

### Conclusion
The final answer is:
\[ \boxed{0} \]