If `|z| = z + 3 - 2i`, then z equals

Options

A. `7//6 + i`
B. `-7//6 + 2i`
C. `-5//6 + 2i`
D. `5//6 + i`

Answer

To solve the equation \( |z| = z + 3 - 2i \), we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers.

### Step 2: Write the modulus of \( z \)
The modulus of \( z \) is given by:
\[ |z| = \sqrt{a^2 + b^2} \]

### Step 3: Substitute \( z \) into the given equation
Substituting \( z \) into the equation \( |z| = z + 3 - 2i \), we have:
\[ \sqrt{a^2 + b^2} = (a + bi) + 3 - 2i \]
This simplifies to:
\[ \sqrt{a^2 + b^2} = (a + 3) + (b - 2)i \]

### Step 4: Separate real and imaginary parts
Now, we can separate the real and imaginary parts:
- The real part: \( \sqrt{a^2 + b^2} = a + 3 \)
- The imaginary part: \( 0 = b - 2 \)

### Step 5: Solve the imaginary part
From the imaginary part, we find:
\[ b - 2 = 0 \implies b = 2 \]

### Step 6: Substitute \( b \) into the real part
Now substitute \( b = 2 \) into the real part:
\[ \sqrt{a^2 + 2^2} = a + 3 \]
This simplifies to:
\[ \sqrt{a^2 + 4} = a + 3 \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ a^2 + 4 = (a + 3)^2 \]
Expanding the right side:
\[ a^2 + 4 = a^2 + 6a + 9 \]

### Step 8: Simplify the equation
Subtract \( a^2 \) from both sides:
\[ 4 = 6a + 9 \]
Rearranging gives:
\[ 6a = 4 - 9 \implies 6a = -5 \implies a = -\frac{5}{6} \]

### Step 9: Write the final result
Now we can write \( z \):
\[ z = a + bi = -\frac{5}{6} + 2i \]

Thus, the solution is:
\[ z = -\frac{5}{6} + 2i \]