If `z_(1)` and `z_(2)` are to complex numbers such that two `|z_(1)|=|z_(2)|+|z_(1)-z_(2)|`, then arg `(z_(1))-"arg"(z_(2))`

Options

A. 0
B. `pi//2`
C. `-pi//2`
D. none of these

Explanation

To solve the problem, we start with the given equation involving the complex numbers \( z_1 \) and \( z_2 \):

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

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate the modulus terms:

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

### Step 2: Squaring Both Sides
Next, we square both sides of the equation to eliminate the absolute values:

\[ (|z_1| - |z_2|)^2 = |z_1 - z_2|^2 \]

### Step 3: Expanding Both Sides
Now, we expand both sides using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):

\[ |z_1|^2 - 2|z_1||z_2| + |z_2|^2 = |z_1|^2 + |z_2|^2 - 2|z_1||z_2|\cos(\theta_1 - \theta_2) \]

where \( \theta_1 = \arg(z_1) \) and \( \theta_2 = \arg(z_2) \).

### Step 4: Canceling Common Terms
We can cancel \( |z_1|^2 + |z_2|^2 \) from both sides:

\[ -2|z_1||z_2| = -2|z_1||z_2|\cos(\theta_1 - \theta_2) \]

### Step 5: Simplifying the Equation
Dividing both sides by \(-2|z_1||z_2|\) (assuming \( |z_1| \) and \( |z_2| \) are not zero):

\[ 1 = \cos(\theta_1 - \theta_2) \]

### Step 6: Finding the Argument Difference
The equation \( \cos(\theta_1 - \theta_2) = 1 \) implies:

\[ \theta_1 - \theta_2 = 0 \]

Thus, we conclude that:

\[ \arg(z_1) - \arg(z_2) = 0 \]

### Final Answer
The difference in arguments is:

\[ \arg(z_1) - \arg(z_2) = 0 \]