Number of imaginary complex numbers satisfying the equation, `z^2=bar(z)2^(1-|z|)` is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To solve the equation \( z^2 = \overline{z} \cdot 2^{1 - |z|} \) for the number of imaginary complex numbers, we will follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ z^2 = \overline{z} \cdot 2^{1 - |z|} \]

2. **Multiply Both Sides by \( z \)**:
Multiply both sides by \( z \):
\[ z^3 = z \cdot \overline{z} \cdot 2^{1 - |z|} \]
Here, \( z \cdot \overline{z} = |z|^2 \), so we can rewrite it as:
\[ z^3 = |z|^2 \cdot 2^{1 - |z|} \]

3. **Take the Modulus of Both Sides**:
Taking the modulus of both sides gives:
\[ |z^3| = ||z|^2 \cdot 2^{1 - |z|}| \]
Using the property of modulus, this simplifies to:
\[ |z|^3 = |z|^2 \cdot 2^{1 - |z|} \]

4. **Assuming \( |z| \neq 0 \)**:
We can divide both sides by \( |z|^2 \) (since \( |z| \neq 0 \)):
\[ |z| = 2^{1 - |z|} \]

5. **Let \( r = |z| \)**:
Substitute \( r \) for \( |z| \):
\[ r = 2^{1 - r} \]

6. **Solve for \( r \)**:
To solve this equation, we can use trial and error or graphical methods. Let's try \( r = 1 \):
\[ 1 = 2^{1 - 1} = 2^0 = 1 \]
This works, so \( r = 1 \) is a solution.

7. **Check for Other Solutions**:
To check if there are other solutions, we can analyze the function \( f(r) = r - 2^{1 - r} \). The function \( f(r) \) is continuous and can be analyzed for intersections with the x-axis.

8. **Finding Roots**:
By checking values:
- For \( r = 0 \), \( f(0) = 0 - 2^1 = -2 \)
- For \( r = 2 \), \( f(2) = 2 - 2^{-1} = 2 - 0.5 = 1.5 \)

Since \( f(0) < 0 \) and \( f(2) > 0 \), there is a root in the interval \( (0, 2) \). However, through graphical or numerical methods, we find that the only positive solution is \( r = 1 \).

9. **Finding \( z \)**:
Since \( |z| = 1 \), we can express \( z \) in terms of its imaginary component:
\[ z = e^{i\theta} \quad \text{for } \theta \in [0, 2\pi) \]

10. **Imaginary Complex Numbers**:
The imaginary complex numbers are those where \( z = i \) or \( z = -i \). The solutions \( z = e^{i\frac{\pi}{2}} \) and \( z = e^{i\frac{3\pi}{2}} \) correspond to these two points.

### Conclusion:
Thus, the number of imaginary complex numbers satisfying the equation is **2**.

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