Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)| + |z_(2)|^(2)`
One of the possible argument of complex number `i(z_(1)//z_(2))`

Options

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

Explanation

To solve the problem, we start with 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: Use the property of modulus
We know that for any complex numbers \( z_1 \) and \( z_2 \):

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

Given that \( |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \), we can set the two expressions equal to each other:

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

### Step 2: Simplify the equation
By subtracting \( |z_1|^2 + |z_2|^2 \) from both sides, we have:

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

### Step 3: Rearranging the terms
This implies:

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

### Step 4: Divide both sides
Dividing both sides by \( z_2 \) (assuming \( z_2 \neq 0 \)) gives:

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

### Step 5: Let \( z = \frac{z_1}{z_2} \)
Let \( z = \frac{z_1}{z_2} \). Then we have:

\[ z = -\overline{z} \]

### Step 6: Analyze the equation
This means that:

\[ z + \overline{z} = 0 \]

This implies that the real part of \( z \) is zero, which means \( z \) is purely imaginary.

### Step 7: Write \( z \) in terms of imaginary unit
Let \( z = i b \) for some real number \( b \). Then we have:

\[ i \frac{z_1}{z_2} = i(i b) = -b \]

### Step 8: Determine the argument
Since \( -b \) is a real number, the argument of a purely real number is either \( 0 \) (if \( b > 0 \)) or \( \pi \) (if \( b < 0 \)). Thus, the possible arguments of \( i \frac{z_1}{z_2} \) can be:

\[ \text{arg}(i \frac{z_1}{z_2}) = \begin{cases} 0 & \text{if } b > 0 \\ \pi & \text{if } b < 0 \end{cases} \]

### Conclusion
Thus, one of the possible arguments of the complex number \( i \frac{z_1}{z_2} \) is \( \pi \).

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