The solution of the equation `|z|-z=1+2i` is

Options

A. `3/2+2i`
B. `3/2-2i`
C. `3-2i`
D. None of these

Answer

To solve the equation \( |z| - z = 1 + 2i \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ |z| - z = 1 + 2i \]
From this, we can express \( z \) in terms of \( |z| \):
\[ z = |z| - (1 + 2i) \]

### Step 2: Express \( z \) and its conjugate
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then, the modulus \( |z| \) can be expressed as:
\[ |z| = \sqrt{x^2 + y^2} \]
Substituting this into our expression for \( z \):
\[ x + yi = \sqrt{x^2 + y^2} - 1 - 2i \]
From this, we can separate the real and imaginary parts:
\[ x = \sqrt{x^2 + y^2} - 1 \quad \text{(1)} \]
\[ y = -2 \quad \text{(2)} \]

### Step 3: Substitute \( y \) into the modulus equation
From equation (2), we have \( y = -2 \). Now substitute \( y \) into the modulus equation:
\[ |z| = \sqrt{x^2 + (-2)^2} = \sqrt{x^2 + 4} \]
Now substitute this back into equation (1):
\[ x = \sqrt{x^2 + 4} - 1 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ x^2 = (\sqrt{x^2 + 4} - 1)^2 \]
Expanding the right-hand side:
\[ x^2 = x^2 + 4 - 2\sqrt{x^2 + 4} + 1 \]
This simplifies to:
\[ 0 = 5 - 2\sqrt{x^2 + 4} \]
Rearranging gives:
\[ 2\sqrt{x^2 + 4} = 5 \]
Dividing by 2:
\[ \sqrt{x^2 + 4} = \frac{5}{2} \]

### Step 5: Square again to solve for \( x \)
Squaring both sides again:
\[ x^2 + 4 = \left(\frac{5}{2}\right)^2 \]
Calculating the right side:
\[ x^2 + 4 = \frac{25}{4} \]
Subtracting 4 (which is \( \frac{16}{4} \)):
\[ x^2 = \frac{25}{4} - \frac{16}{4} = \frac{9}{4} \]
Taking the square root gives:
\[ x = \pm \frac{3}{2} \]

### Step 6: Find the corresponding \( z \)
Now we have two possible values for \( x \):
1. \( x = \frac{3}{2} \)
2. \( x = -\frac{3}{2} \)

Using \( y = -2 \):
- For \( x = \frac{3}{2} \):
\[ z = \frac{3}{2} - 2i \]

- For \( x = -\frac{3}{2} \):
\[ z = -\frac{3}{2} - 2i \]

### Step 7: Check which solution satisfies the original equation
We need to check which of these satisfies the original equation \( |z| - z = 1 + 2i \).

1. For \( z = \frac{3}{2} - 2i \):
- \( |z| = \sqrt{\left(\frac{3}{2}\right)^2 + (-2)^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{25}{4}} = \frac{5}{2} \)
- \( |z| - z = \frac{5}{2} - \left(\frac{3}{2} - 2i\right) = \frac{5}{2} - \frac{3}{2} + 2i = 1 + 2i \) (satisfies)

2. For \( z = -\frac{3}{2} - 2i \):
- \( |z| = \sqrt{\left(-\frac{3}{2}\right)^2 + (-2)^2} = \frac{5}{2} \)
- \( |z| - z = \frac{5}{2} - \left(-\frac{3}{2} - 2i\right) = \frac{5}{2} + \frac{3}{2} + 2i = 4 + 2i \) (does not satisfy)

### Final Answer
Thus, the solution to the equation \( |z| - z = 1 + 2i \) is:
\[ z = \frac{3}{2} - 2i \]