If `alpha` and `beta` are sum and product of non zero solution of the equation `overline (z)^2 + |z| = 0` where z is a complex number then value of `4(alpha^2 + beta^2)` is equal to

Options

A. 2
B. 6
C. 4
D. 8

Answer

To solve the equation \( \overline{z}^2 + |z| = 0 \) where \( z \) is a complex number, we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate of \( z \) is given by:
\[ \overline{z} = x - iy \]

### Step 2: Substitute \( \overline{z} \) and \( |z| \) into the equation
The modulus \( |z| \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Substituting these into the equation:
\[ (x - iy)^2 + \sqrt{x^2 + y^2} = 0 \]

### Step 3: Expand \( (x - iy)^2 \)
Calculating \( (x - iy)^2 \):
\[ (x - iy)^2 = x^2 - 2xyi - y^2 = (x^2 - y^2) - 2xyi \]
Thus, the equation becomes:
\[ (x^2 - y^2) - 2xyi + \sqrt{x^2 + y^2} = 0 \]

### Step 4: Separate the real and imaginary parts
For the equation to hold, both the real part and the imaginary part must equal zero:
1. Real part:
\[ x^2 - y^2 + \sqrt{x^2 + y^2} = 0 \]
2. Imaginary part:
\[ -2xy = 0 \]

### Step 5: Solve the imaginary part
From \( -2xy = 0 \), we have two cases:
- Case 1: \( x = 0 \)
- Case 2: \( y = 0 \)

### Step 6: Analyze each case
**Case 1: \( x = 0 \)**
Substituting \( x = 0 \) into the real part:
\[ 0 - y^2 + \sqrt{0 + y^2} = 0 \implies -y^2 + |y| = 0 \]
This gives:
\[ |y| = y^2 \]
Thus, \( y = 1 \) or \( y = -1 \). Therefore, the solutions are \( z = i \) and \( z = -i \).

**Case 2: \( y = 0 \)**
Substituting \( y = 0 \) into the real part:
\[ x^2 - 0 + |x| = 0 \implies x^2 + |x| = 0 \]
This implies \( x = 0 \) (since \( x \) cannot be negative). Thus, this case does not yield any non-zero solutions.

### Step 7: Identify non-zero solutions
The non-zero solutions from Case 1 are:
- \( z = i \) (where \( x = 0, y = 1 \))
- \( z = -i \) (where \( x = 0, y = -1 \))

### Step 8: Calculate \( \alpha \) and \( \beta \)
Let:
- \( \alpha \) be the sum of the solutions:
\[ \alpha = i + (-i) = 0 \]
- \( \beta \) be the product of the solutions:
\[ \beta = i \cdot (-i) = 1 \]

### Step 9: Calculate \( 4(\alpha^2 + \beta^2) \)
Now, we compute:
\[ \alpha^2 = 0^2 = 0 \]
\[ \beta^2 = 1^2 = 1 \]
Thus,
\[ 4(\alpha^2 + \beta^2) = 4(0 + 1) = 4 \]

### Final Answer
The value of \( 4(\alpha^2 + \beta^2) \) is \( \boxed{4} \).