If `k + |k + z^2|=|z|^2(k in R^-)`, then possible argument of z is

Options

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

Explanation

To solve the equation \( k + |k + z^2| = |z|^2 \) where \( k \in \mathbb{R}^- \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ k + |k + z^2| = |z|^2 \]

### Step 2: Express \( z \) in terms of its imaginary part
Let \( z = ni \), where \( n \) is a real number. Then, we have:
\[ z^2 = (ni)^2 = -n^2 \]
Thus, the equation becomes:
\[ k + |k - n^2| = |ni|^2 \]

### Step 3: Calculate \( |z|^2 \)
The modulus squared of \( z \) is:
\[ |z|^2 = |ni|^2 = n^2 \]
Now, substituting this into our equation gives:
\[ k + |k - n^2| = n^2 \]

### Step 4: Analyze the modulus
Since \( k \) is negative, we need to consider two cases for \( |k - n^2| \):

#### Case 1: \( k - n^2 \geq 0 \) (which is not possible since \( k < 0 \))
This case does not apply because \( k \) is negative.

#### Case 2: \( k - n^2 < 0 \)
In this case:
\[ |k - n^2| = -(k - n^2) = n^2 - k \]
Substituting this back into the equation gives:
\[ k + (n^2 - k) = n^2 \]
This simplifies to:
\[ n^2 = n^2 \]
This is always true.

### Step 5: Determine the argument of \( z \)
Since \( z = ni \), the argument of \( z \) is:
\[ \text{arg}(z) = \frac{\pi}{2} \]
This is because \( z \) lies on the positive imaginary axis.

### Conclusion
The possible argument of \( z \) is:
\[ \frac{\pi}{2} \]