The number of solution of the equation `z^2+absz^2`=0,where z is complex number is :

Options

A. 1
B. 2
C. 4
D. infinitely many

Answer

To solve the equation \( z^2 + |z|^2 = 0 \) 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, and \( i \) is the imaginary unit.

### Step 2: Calculate \( z^2 \) and \( |z|^2 \)
We calculate \( z^2 \) and \( |z|^2 \):
- \( z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \)
- \( |z|^2 = x^2 + y^2 \)

### Step 3: Substitute into the equation
Now substitute \( z^2 \) and \( |z|^2 \) into the equation:
\[ (x^2 - y^2) + 2xyi + (x^2 + y^2) = 0 \]

### Step 4: Separate real and imaginary parts
This gives us:
\[ (x^2 - y^2 + x^2 + y^2) + 2xyi = 0 \]
This simplifies to:
\[ (2x^2) + 2xyi = 0 \]

### Step 5: Set real and imaginary parts to zero
For the equation to hold, both the real and imaginary parts must be zero:
1. \( 2x^2 = 0 \)
2. \( 2xy = 0 \)

### Step 6: Solve the equations
From \( 2x^2 = 0 \):
- This implies \( x = 0 \).

From \( 2xy = 0 \):
- This is satisfied if either \( x = 0 \) or \( y = 0 \). Since we already have \( x = 0 \), \( y \) can take any real value.

### Conclusion
Thus, the solutions are of the form \( z = 0 + iy \) where \( y \) can be any real number. Therefore, there are infinitely many solutions.

### Final Answer
The number of solutions of the equation \( z^2 + |z|^2 = 0 \) is **infinitely many**.

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