If `|z_(1)|=|z_(2)|andamp(z_(1))+amp(z_(2))=0,` then

Options

A. `z_(1)=z_(2)`
B. `barz_(1)=z_(2)`
C. `z_(1)+z_(2)=-0`
D. `barz_(1)=barz_(2)`

Answer

To solve the problem, we start with the given conditions:

1. \( |z_1| = |z_2| \)
2. \( \text{amp}(z_1) + \text{amp}(z_2) = 0 \)

We need to analyze these conditions to derive a conclusion about \( z_1 \) and \( z_2 \).

### Step 1: Express \( z_1 \) and \( z_2 \) in exponential form
Since \( z_1 \) and \( z_2 \) are complex numbers, we can express them in polar form:
\[ z_1 = |z_1| e^{i \theta_1} \]
\[ z_2 = |z_2| e^{i \theta_2} \]

### Step 2: Use the modulus condition
From the first condition \( |z_1| = |z_2| \), we can denote this common modulus as \( r \):
\[ z_1 = r e^{i \theta_1} \]
\[ z_2 = r e^{i \theta_2} \]

### Step 3: Use the amplitude condition
From the second condition \( \text{amp}(z_1) + \text{amp}(z_2) = 0 \), we can express this in terms of the angles:
\[ \theta_1 + \theta_2 = 0 \]
This implies:
\[ \theta_2 = -\theta_1 \]

### Step 4: Substitute \( \theta_2 \) into the expression for \( z_2 \)
Substituting \( \theta_2 \) into the expression for \( z_2 \):
\[ z_2 = r e^{-i \theta_1} \]

### Step 5: Relate \( z_2 \) to \( z_1 \)
Now we can express \( z_2 \) in terms of \( z_1 \):
\[ z_2 = r e^{-i \theta_1} = \overline{z_1} \]
This shows that \( z_2 \) is the complex conjugate of \( z_1 \).

### Conclusion
Thus, we conclude that:
\[ z_2 = \overline{z_1} \]
This means that if the modulus of \( z_1 \) and \( z_2 \) are equal and their amplitudes sum to zero, then \( z_2 \) is the complex conjugate of \( z_1 \).

### Final Answer
Therefore, the correct statement is:
\[ z_2 = \overline{z_1} \]