Let `z_(1),z_(2)` be two distinct complex numbers with non-zero real and imaginary parts such that `"arg"(z_(1)+z_(2))=pi//2`, then `"arg"(z_(1)+bar(z)_(1))-"arg"(z_(2)+bar(z)_(2))` is equal to

Options

A. `pi//2`
B. `pi`
C. `-pi//2`
D. None of these.

Answer

To solve the problem, we need to analyze the given conditions and derive the required expression step by step.

### Step 1: Define the Complex Numbers
Let \( z_1 = x_1 + i y_1 \) and \( z_2 = x_2 + i y_2 \), where \( x_1, y_1, x_2, y_2 \) are real numbers and \( x_1, y_1, x_2, y_2 \neq 0 \).

### Step 2: Use the Given Condition
We know that:
\[ \text{arg}(z_1 + z_2) = \frac{\pi}{2} \]
This implies that the real part of \( z_1 + z_2 \) must be zero because the argument of a complex number is \( \frac{\pi}{2} \) when it lies on the imaginary axis. Therefore:
\[ x_1 + x_2 = 0 \implies x_2 = -x_1 \]

### Step 3: Express \( z_2 \)
Substituting \( x_2 \) in \( z_2 \):
\[ z_2 = -x_1 + i y_2 \]

### Step 4: Calculate \( z_1 + \bar{z_1} \)
Now, we compute \( z_1 + \bar{z_1} \):
\[ \bar{z_1} = x_1 - i y_1 \]
Thus,
\[ z_1 + \bar{z_1} = (x_1 + i y_1) + (x_1 - i y_1) = 2x_1 \]
The argument of a real number \( 2x_1 \) (which is positive since \( x_1 \neq 0 \)) is:
\[ \text{arg}(z_1 + \bar{z_1}) = 0 \]

### Step 5: Calculate \( z_2 + \bar{z_2} \)
Next, we compute \( z_2 + \bar{z_2} \):
\[ \bar{z_2} = -x_1 - i y_2 \]
Thus,
\[ z_2 + \bar{z_2} = (-x_1 + i y_2) + (-x_1 - i y_2) = -2x_1 \]
The argument of \( -2x_1 \) (which is negative since \( x_1 \neq 0 \)) is:
\[ \text{arg}(z_2 + \bar{z_2}) = \pi \]

### Step 6: Find the Required Expression
Now we need to find:
\[ \text{arg}(z_1 + \bar{z_1}) - \text{arg}(z_2 + \bar{z_2}) \]
Substituting the values we found:
\[ 0 - \pi = -\pi \]

### Conclusion
The final result is:
\[ \text{arg}(z_1 + \bar{z_1}) - \text{arg}(z_2 + \bar{z_2}) = -\pi \]