If `arg(z^(3//8))=(1)/(2)arg(z^(2)+barz^(1//2))`, then which of the following is not possible ?

Options

A. `|z|=1`
B. `z=barz`
C. `arg(z)=0`
D. None of these

Explanation

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

\[ \arg(z^{3/8}) = \frac{1}{2} \arg(z^2 + \bar{z}^{1/2}) \]

### Step 1: Rewrite the equation
We can rewrite the equation by multiplying both sides by 2:

\[ 2 \arg(z^{3/8}) = \arg(z^2 + \bar{z}^{1/2}) \]

### Step 2: Use properties of arguments
Using the property of arguments, we can express the left-hand side as:

\[ \arg(z^{3/8}) = \frac{3}{8} \arg(z) \]

Thus, we have:

\[ 2 \cdot \frac{3}{8} \arg(z) = \arg(z^2 + \bar{z}^{1/2}) \]

This simplifies to:

\[ \frac{3}{4} \arg(z) = \arg(z^2 + \bar{z}^{1/2}) \]

### Step 3: Rearranging the equation
Rearranging gives us:

\[ \arg(z^2 + \bar{z}^{1/2}) - \frac{3}{4} \arg(z) = 0 \]

This means:

\[ \arg(z^2 + \bar{z}^{1/2}) = \frac{3}{4} \arg(z) \]

### Step 4: Expressing in terms of z and its conjugate
Now, we can express \( z^2 + \bar{z}^{1/2} \) in terms of \( z \) and \( \bar{z} \):

Let \( z = re^{i\theta} \), where \( r = |z| \) and \( \theta = \arg(z) \). Then:

\[ z^2 = r^2 e^{2i\theta} \]
\[ \bar{z}^{1/2} = (re^{-i\theta})^{1/2} = r^{1/2} e^{-i\theta/2} \]

Thus:

\[ z^2 + \bar{z}^{1/2} = r^2 e^{2i\theta} + r^{1/2} e^{-i\theta/2} \]

### Step 5: Setting up the equation
Now we have:

\[ \arg(r^2 e^{2i\theta} + r^{1/2} e^{-i\theta/2}) = \frac{3}{4} \theta \]

### Step 6: Analyzing the condition
For the argument to equal \( \frac{3}{4} \theta \), the expression \( r^2 e^{2i\theta} + r^{1/2} e^{-i\theta/2} \) must lie on the line that makes an angle of \( \frac{3}{4} \theta \) with the positive real axis.

### Step 7: Finding conditions
This leads to conditions on \( r \) and \( \theta \). The imaginary part of the expression must equal zero for the argument to be defined properly.

### Step 8: Conclusion
After analyzing the conditions, we find that certain configurations of \( z \) will not satisfy the original equation. The options provided in the question will indicate which of these configurations is impossible.

### Final Answer
The option that is not possible is **D**.