Complex numbers z satisfy the equaiton `|z-(4//z)|=2`
The value of `arg(z_(1)//z_(2))` where `z_(1)` and `z_(2)` are complex numbers with the greatest and the least moduli, can be

Options

A. (a) `2pi`
B. (b) `pi`
C. (c) `pi//2`
D. (d) none of these

Explanation

To solve the problem, we start with the equation given for the complex number \( z \):

\[ |z - \frac{4}{z}| = 2 \]

### Step 1: Rewrite the equation
We can rewrite the equation using the property of modulus. Let \( z = re^{i\theta} \), where \( r = |z| \) and \( \theta = \arg(z) \). Then, we have:

\[ |z - \frac{4}{z}| = |z - 4 \cdot \frac{1}{z}| = |z - 4 \cdot \frac{1}{re^{i\theta}}| = |re^{i\theta} - \frac{4}{r} e^{-i\theta}| \]

### Step 2: Square both sides
Squaring both sides gives:

\[ |z - \frac{4}{z}|^2 = 4 \]

This leads to:

\[ (z - \frac{4}{z})(\overline{z} - \frac{4}{\overline{z}}) = 4 \]

Substituting \( z = re^{i\theta} \) and \( \overline{z} = re^{-i\theta} \):

\[ (re^{i\theta} - \frac{4}{r} e^{-i\theta})(re^{-i\theta} - \frac{4}{r} e^{i\theta}) = 4 \]

### Step 3: Expand the expression
Expanding the left-hand side:

\[ \left(r^2 - \frac{4}{r} e^{i\theta} \cdot re^{-i\theta} - \frac{4}{r} re^{i\theta} \cdot e^{-i\theta} + \frac{16}{r^2}\right) = 4 \]

This simplifies to:

\[ r^2 + \frac{16}{r^2} - 4 = 4 \]

### Step 4: Rearranging the equation
Rearranging gives:

\[ r^2 + \frac{16}{r^2} = 8 \]

### Step 5: Multiply through by \( r^2 \)
Multiplying through by \( r^2 \) gives:

\[ r^4 - 8r^2 + 16 = 0 \]

### Step 6: Let \( x = r^2 \)
Let \( x = r^2 \). The equation becomes:

\[ x^2 - 8x + 16 = 0 \]

### Step 7: Factor the quadratic
Factoring gives:

\[ (x - 4)^2 = 0 \]

Thus, \( x = 4 \) which implies:

\[ r^2 = 4 \implies r = 2 \]

### Step 8: Finding the argument
Now, we need to find the values of \( \arg(z_1/z_2) \) where \( z_1 \) and \( z_2 \) are complex numbers with the greatest and least moduli. Since both \( z_1 \) and \( z_2 \) have the same modulus \( r = 2 \), we can express them as:

\[ z_1 = 2e^{i\theta_1}, \quad z_2 = 2e^{i\theta_2} \]

Thus:

\[ \frac{z_1}{z_2} = \frac{2e^{i\theta_1}}{2e^{i\theta_2}} = e^{i(\theta_1 - \theta_2)} \]

### Step 9: Finding the argument
The argument of \( z_1/z_2 \) is:

\[ \arg\left(\frac{z_1}{z_2}\right) = \theta_1 - \theta_2 \]

Given that \( \theta_1 \) and \( \theta_2 \) can take values such that the maximum and minimum moduli are achieved, we find that:

\[ \arg(z_1/z_2) = n\pi \quad \text{(where \( n \) is an integer)} \]

### Conclusion
Thus, the possible values of \( \arg(z_1/z_2) \) can be \( \pm \pi \).

### Final Answer
The value of \( \arg(z_1/z_2) \) can be \( \pi \) or \( -\pi \).

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