If a complex number z satisfies `z + sqrt(2) |z + 1| + i=0`, then |z| is equal to :

Options

A. 2
B. `sqrt(3)`
C. `sqrt(5)`
D. 1

Answer

To solve the problem, we need to find the modulus of the complex number \( z \) that satisfies the equation:

\[ z + \sqrt{2} |z + 1| + i = 0 \]

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can rewrite the equation as:

\[ (x + iy) + \sqrt{2} |(x + 1) + iy| + i = 0 \]

### Step 2: Calculate \( |z + 1| \)
Now, calculate \( |z + 1| \):

\[ z + 1 = (x + 1) + iy \]

The modulus is given by:

\[ |z + 1| = \sqrt{(x + 1)^2 + y^2} \]

### Step 3: Substitute back into the equation
Substituting \( |z + 1| \) back into the equation gives:

\[ x + iy + \sqrt{2} \sqrt{(x + 1)^2 + y^2} + i = 0 \]

### Step 4: Separate real and imaginary parts
Now, separate the real and imaginary parts:

Real part:
\[ x + \sqrt{2} \sqrt{(x + 1)^2 + y^2} = 0 \]

Imaginary part:
\[ y + 1 = 0 \]

### Step 5: Solve the imaginary part
From the imaginary part, we find:

\[ y + 1 = 0 \implies y = -1 \]

### Step 6: Substitute \( y \) into the real part
Now substitute \( y = -1 \) into the real part:

\[ x + \sqrt{2} \sqrt{(x + 1)^2 + (-1)^2} = 0 \]

This simplifies to:

\[ x + \sqrt{2} \sqrt{(x + 1)^2 + 1} = 0 \]

### Step 7: Simplify the equation
Squaring both sides to eliminate the square root gives:

\[ x + \sqrt{2} \sqrt{(x + 1)^2 + 1} = 0 \]

Rearranging gives:

\[ \sqrt{2} \sqrt{(x + 1)^2 + 1} = -x \]

Squaring both sides:

\[ 2((x + 1)^2 + 1) = x^2 \]

Expanding and simplifying:

\[ 2(x^2 + 2x + 2) = x^2 \]
\[ 2x^2 + 4x + 4 = x^2 \]
\[ x^2 + 4x + 4 = 0 \]

### Step 8: Solve the quadratic equation
Factoring gives:

\[ (x + 2)^2 = 0 \implies x = -2 \]

### Step 9: Find \( z \)
Now we have \( x = -2 \) and \( y = -1 \), so:

\[ z = -2 - i \]

### Step 10: Calculate the modulus of \( z \)
Finally, the modulus of \( z \) is:

\[ |z| = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

Thus, the answer is:

\[ |z| = \sqrt{5} \]