For any two complex numbers `z_1` and `z_2`, we have `|z_1+z_2|^2=|z_1|^2+|z_2|^2`, then

Options

A. `Re((z_(1))/(z_(2)))=0`
B. `Im((z_(1))/(z_(2)))=0`
C. `Re(z_(1)z_(2))=0 `
D. `Im (z_(1)z_(2))=0 `

Answer

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

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

### Step 1: Expand the left-hand side
We start by expanding the left-hand side using the property of modulus:
\[ |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = (z_1 + z_2)(\overline{z_1} + \overline{z_2}) \]
This expands to:
\[ |z_1 + z_2|^2 = z_1\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_1} + z_2\overline{z_2} \]
Using the fact that \( |z|^2 = z\overline{z} \), we can rewrite this as:
\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + z_2\overline{z_1} \]

### Step 2: Set the equation
Now we 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: Simplify the equation
Subtract \( |z_1|^2 + |z_2|^2 \) from both sides:
\[ z_1\overline{z_2} + z_2\overline{z_1} = 0 \]

### Step 4: Rearranging the equation
This can be rearranged as:
\[ z_1\overline{z_2} = -z_2\overline{z_1} \]

### Step 5: Divide by \( z_2 \) (assuming \( z_2 \neq 0 \))
Dividing both sides by \( z_2 \) (assuming \( z_2 \neq 0 \)):
\[ \frac{z_1}{z_2} = -\frac{\overline{z_1}}{\overline{z_2}} \]

### Step 6: Conclude that the quotient is purely imaginary
This implies that the quotient \( \frac{z_1}{z_2} \) is purely imaginary, meaning that the real part of \( \frac{z_1}{z_2} \) is zero.

### Final Conclusion
Thus, we conclude that:
\[ \text{Re}\left(\frac{z_1}{z_2}\right) = 0 \]