If ` z_(1) and z_(2)` be two complex numbers such that `| z_(1) - z_(2)| = | z_(1)| - | z_(2)| ` , then arg ` (z_(1))/( z_(2))` is

Options

A. `pi//4`
B. `pi//2`
C. 0
D. none

Answer

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

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

### Step 1: Square both sides
We start by squaring both sides of the equation:
\[ |z_1 - z_2|^2 = (|z_1| - |z_2|)^2 \]

### Step 2: Expand both sides
Using the property of modulus, we can expand both sides:
\[ |z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \overline{z_2}) \]
And for the right side:
\[ (|z_1| - |z_2|)^2 = |z_1|^2 - 2|z_1||z_2| + |z_2|^2 \]

### Step 3: Set the two expansions equal
Now we equate the two expressions:
\[ |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \overline{z_2}) = |z_1|^2 - 2|z_1||z_2| + |z_2|^2 \]

### Step 4: Simplify the equation
Cancel \( |z_1|^2 \) and \( |z_2|^2 \) from both sides:
\[ -2 \text{Re}(z_1 \overline{z_2}) = -2|z_1||z_2| \]
Dividing both sides by -2 gives:
\[ \text{Re}(z_1 \overline{z_2}) = |z_1||z_2| \]

### Step 5: Analyze the result
The equation \( \text{Re}(z_1 \overline{z_2}) = |z_1||z_2| \) indicates that the angle between \( z_1 \) and \( z_2 \) is zero. This means that \( z_1 \) and \( z_2 \) are in the same direction.

### Step 6: Conclusion about the argument
Since \( z_1 \) and \( z_2 \) are in the same direction, we have:
\[ \arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) = 0 \]

Thus, the final answer is:
\[ \arg\left(\frac{z_1}{z_2}\right) = 0 \]