If `|z+1| = z+ 2 ( 1+ i )`, then the value of z is

Options

A. `(1)/( 2) - 2i`
B. `( 1)/( 2) + 2i`
C. `( 1)/( 2) - 3i`
D. `(1)/( 3) - 2i`

Answer

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

### Step 1: Let \( z = x + iy \)
We express the complex number \( z \) in terms of its real part \( x \) and imaginary part \( y \).

### Step 2: Rewrite the equation
Substituting \( z \) into the equation, we have:
\[ |z + 1| = |(x + 1) + iy| = \sqrt{(x + 1)^2 + y^2} \]
And the right side becomes:
\[ z + 2(1 + i) = (x + iy) + 2 + 2i = (x + 2) + (y + 2)i \]

### Step 3: Set up the equation
Now we can rewrite the equation as:
\[ \sqrt{(x + 1)^2 + y^2} = (x + 2) + (y + 2)i \]
Since the left side is a magnitude (which is a real number), the imaginary part on the right must equal zero:
\[ y + 2 = 0 \]

### Step 4: Solve for \( y \)
From \( y + 2 = 0 \), we find:
\[ y = -2 \]

### Step 5: Substitute \( y \) back into the equation
Now substituting \( y = -2 \) into the equation for the magnitudes:
\[ \sqrt{(x + 1)^2 + (-2)^2} = x + 2 \]
This simplifies to:
\[ \sqrt{(x + 1)^2 + 4} = x + 2 \]

### Step 6: Square both sides
Squaring both sides gives:
\[ (x + 1)^2 + 4 = (x + 2)^2 \]

### Step 7: Expand both sides
Expanding both sides results in:
\[ x^2 + 2x + 1 + 4 = x^2 + 4x + 4 \]
This simplifies to:
\[ x^2 + 2x + 5 = x^2 + 4x + 4 \]

### Step 8: Rearrange the equation
Subtract \( x^2 \) from both sides:
\[ 2x + 5 = 4x + 4 \]
Rearranging gives:
\[ 5 - 4 = 4x - 2x \]
This simplifies to:
\[ 1 = 2x \]

### Step 9: Solve for \( x \)
Dividing both sides by 2 gives:
\[ x = \frac{1}{2} \]

### Step 10: Write the final value of \( z \)
Now substituting \( x \) and \( y \) back into the expression for \( z \):
\[ z = x + iy = \frac{1}{2} - 2i \]

### Final Answer
The value of \( z \) is:
\[ \boxed{\frac{1}{2} - 2i} \]