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`

Answer

To solve the problem, we need to find the modulus of the complex number \( z \) given the equation \( |z| + z = 3 + i \).

### Step-by-Step Solution:

1. **Express \( z \) in terms of its components**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i = \sqrt{-1} \).

2. **Write the modulus of \( z \)**:
The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

3. **Substitute \( z \) and \( |z| \) into the equation**:
Substitute \( z \) and its modulus into the equation:
\[ \sqrt{x^2 + y^2} + (x + iy) = 3 + i \]

4. **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: \( y = 1 \)

5. **Substitute \( y \) into the real part equation**:
Substitute \( y = 1 \) into the real part equation:
\[ \sqrt{x^2 + 1^2} + x = 3 \]
This simplifies to:
\[ \sqrt{x^2 + 1} + x = 3 \]

6. **Isolate the square root**:
Rearranging gives:
\[ \sqrt{x^2 + 1} = 3 - x \]

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

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

9. **Find \( y \)**:
We already found \( y = 1 \).

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

### Final Answer:
Thus, the modulus \( |z| \) is equal to \( \frac{5}{3} \).

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