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`

Explanation

To solve the problem, we need to analyze the equation given for two 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:

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

Expanding this, we get:

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

### Step 2: Substitute the modulus

We know that \( |z_1|^2 = z_1 \overline{z_1} \) and \( |z_2|^2 = z_2 \overline{z_2} \). Thus, we can rewrite the equation as:

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

### Step 3: Set the equation

Now, substituting back into the equation we have:

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

### Step 4: Simplify the equation

Subtracting \( |z_1|^2 + |z_2|^2 \) from both sides gives us:

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

### Step 5: Analyze the result

The expression \( z_1 \overline{z_2} + z_2 \overline{z_1} = 0 \) implies that:

\[ \text{Re}(z_1 \overline{z_2}) = 0 \]

This means that the product \( z_1 \overline{z_2} \) is purely imaginary.

### Step 6: Relate to angles

Let \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \). Then:

\[ z_1 \overline{z_2} = r_1 r_2 e^{i(\theta_1 - \theta_2)} \]

For this to be purely imaginary, the angle \( \theta_1 - \theta_2 \) must be \( \frac{\pi}{2} + k\pi \) for some integer \( k \). The simplest case is when:

\[ \theta_1 - \theta_2 = \frac{\pi}{2} \]

### Step 7: Conclusion

This means that the arguments of \( z_1 \) and \( z_2 \) differ by \( \frac{\pi}{2} \), which indicates that \( z_1 \) and \( z_2 \) are orthogonal in the complex plane.

Thus, the conclusion is that if \( |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \), then the complex numbers \( z_1 \) and \( z_2 \) are orthogonal.

### Final Answer

The correct option is **A**: \( z_1 \) and \( z_2 \) are orthogonal.

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