If `z_1` and `z_2` are two nonzero complex numbers such that `|z_1-z_2|=|z_1|-|z_2|` then arg `z_1`-arg`z_2` is equal to

Options

A. 0
B. `pi`
C. `pi/2`
D. `-pi/2`

Answer

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

\[ |z_1 - z_2| = |z_1| - |z_2| \]

### Step 1: Understanding the Condition
The equation \( |z_1 - z_2| = |z_1| - |z_2| \) suggests a geometric interpretation. The left-hand side represents the distance between the points \( z_1 \) and \( z_2 \) in the complex plane, while the right-hand side represents the difference in their magnitudes.

### Step 2: Applying the Triangle Inequality
From the properties of complex numbers, we know that:

\[ |z_1| - |z_2| \leq |z_1 - z_2| \]

This inequality holds for any complex numbers \( z_1 \) and \( z_2 \). The equality \( |z_1 - z_2| = |z_1| - |z_2| \) can only occur when \( z_1 \) and \( z_2 \) are collinear and point in the same direction.

### Step 3: Geometric Interpretation
If \( z_1 \) and \( z_2 \) are collinear, they can be represented as:

\[ z_1 = r_1 e^{i\theta_1}, \quad z_2 = r_2 e^{i\theta_2} \]

where \( r_1 = |z_1| \), \( r_2 = |z_2| \), and \( \theta_1 = \text{arg}(z_1) \), \( \theta_2 = \text{arg}(z_2) \).

### Step 4: Finding the Argument Difference
Since \( z_1 \) and \( z_2 \) are on the same line, the angles \( \theta_1 \) and \( \theta_2 \) must be equal or differ by \( \pi \) (180 degrees). Therefore, we can conclude:

\[ \text{arg}(z_1) - \text{arg}(z_2) = 0 \quad \text{or} \quad \text{arg}(z_1) - \text{arg}(z_2) = \pi \]

### Step 5: Conclusion
However, since both \( z_1 \) and \( z_2 \) are nonzero complex numbers, the case where they differ by \( \pi \) would imply that one is the negative of the other, which would not satisfy the original equation. Thus, we conclude:

\[ \text{arg}(z_1) - \text{arg}(z_2) = 0 \]

### Final Answer
Therefore, the answer is:

\[ \text{arg}(z_1) - \text{arg}(z_2) = 0 \]

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