Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)|^2 + |z_(2)|^(2)` Complex number `z_(1)/z_(2)` is

Options

A. purely real
B. purely imaginary
C. zero
D. none of theses

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:

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

This expands to:

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

Since \( |z_1|^2 = z_1\overline{z_1} \) and \( |z_2|^2 = z_2\overline{z_2} \), 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 expanded form equal to the right-hand side
Now, we set this equal to the right-hand side of the original equation:

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

### 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
We can rearrange this to:

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

### Step 5: Divide both sides by \( z_2\overline{z_2} \)
Assuming \( z_2 \neq 0 \), we divide both sides by \( z_2\overline{z_2} \):

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

### Step 6: Use the property of conjugates
This can be rewritten using the property of conjugates:

\[ \frac{z_1}{z_2} + \frac{z_1}{z_2}\bigg|^* = 0 \]

This indicates that \( \frac{z_1}{z_2} \) is purely imaginary.

### Conclusion
Thus, we conclude that the complex number \( \frac{z_1}{z_2} \) is purely imaginary.