Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)|^(2) + |z_(2)|^(2)` Possible difference between the argument of `z_(1)` and `z_(2)` is

Options

A. 0
B. `pi`
C. `-(pi)/(2)`
D. none of these

Explanation

To solve the problem, we need to analyze the given relation involving the complex numbers \( z_1 \) and \( z_2 \):

\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \]

### Step 1: Expand the left-hand side
Using the property of modulus, we can expand the left-hand side:

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

### Step 2: Set up the equation
Now, we can set the expanded left-hand side equal to the right-hand side:

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

### Step 3: Cancel common terms
By canceling \( |z_1|^2 \) and \( |z_2|^2 \) from both sides, we get:

\[ z_1 \overline{z_2} + z_2 \overline{z_1} = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:

\[ z_1 \overline{z_2} = - z_2 \overline{z_1} \]

### Step 5: Divide both sides
Dividing both sides by \( z_2 \overline{z_2} \) (assuming \( z_2 \neq 0 \)) gives:

\[ \frac{z_1}{z_2} = -\frac{\overline{z_1}}{\overline{z_2}} \]

### Step 6: Use the property of conjugates
This implies:

\[ \frac{z_1}{z_2} = \overline{\left(\frac{z_1}{z_2}\right)} \]

### Step 7: Interpret the result
The equality \( \frac{z_1}{z_2} = \overline{\left(\frac{z_1}{z_2}\right)} \) indicates that \( \frac{z_1}{z_2} \) is a real number. Therefore, the argument of \( z_1 \) and \( z_2 \) must differ by an integer multiple of \( \pi \):

\[ \theta_1 - \theta_2 = n\pi \]

where \( \theta_1 \) and \( \theta_2 \) are the arguments of \( z_1 \) and \( z_2 \), respectively, and \( n \) is an integer.

### Conclusion
Thus, the possible difference between the arguments of \( z_1 \) and \( z_2 \) is:

\[ \theta_1 - \theta_2 = n\pi \quad (n \in \mathbb{Z}) \]