Let z be a complex number such that |z| + z = 3 + i (Where `i=sqrt(-1))` Then ,|z| is equal to

Options

A. `sqrt(34)/3`
B. `5/3`
C. `sqrt(41)/4`
D. `5/4`

Explanation

To solve the problem, we need to find the value of |z| given that |z| + z = 3 + i, where z is a complex number.

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 1: Express |z| in terms of x and y
The modulus of z is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 2: Substitute z and |z| into the equation
Substituting \( z \) and \( |z| \) into the equation |z| + z = 3 + i, we have:
\[ \sqrt{x^2 + y^2} + (x + iy) = 3 + i \]

### Step 3: Separate the real and imaginary parts
From the equation, we can separate the real and imaginary parts:
- Real part: \( \sqrt{x^2 + y^2} + x = 3 \)
- Imaginary part: \( y = 1 \)

### Step 4: Substitute y into the real part equation
Substituting \( y = 1 \) into the real part equation:
\[ \sqrt{x^2 + 1^2} + x = 3 \]
This simplifies to:
\[ \sqrt{x^2 + 1} + x = 3 \]

### Step 5: Isolate the square root
Rearranging gives:
\[ \sqrt{x^2 + 1} = 3 - x \]

### Step 6: Square both sides
Squaring both sides results in:
\[ x^2 + 1 = (3 - x)^2 \]
Expanding the right side:
\[ x^2 + 1 = 9 - 6x + x^2 \]

### Step 7: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 1 = 9 - 6x \]
Rearranging gives:
\[ 6x = 8 \quad \Rightarrow \quad x = \frac{4}{3} \]

### Step 8: Find y
We already found that \( y = 1 \).

### Step 9: Calculate |z|
Now we can find |z|:
\[ |z| = \sqrt{x^2 + y^2} = \sqrt{\left(\frac{4}{3}\right)^2 + 1^2} = \sqrt{\frac{16}{9} + 1} = \sqrt{\frac{16}{9} + \frac{9}{9}} = \sqrt{\frac{25}{9}} = \frac{5}{3} \]

### Conclusion
Thus, the value of |z| is:
\[ |z| = \frac{5}{3} \]