For the complex number z satisfying the condition `|z+(2)/(z)|=2`, the maximum value of `|z|` is

Options

A. `sqrt3-1`
B. `sqrt3+1`
C. `sqrt2+sqrt3`
D. `sqrt3`

Answer

To solve the problem, we need to find the maximum value of \( |z| \) given the condition \( |z + \frac{2}{z}| = 2 \).

### Step-by-Step Solution:

1. **Rewrite the Condition**:
We start with the condition:
\[ |z + \frac{2}{z}| = 2 \]
Let \( z = r e^{i\theta} \) where \( r = |z| \) and \( \theta \) is the argument of \( z \).

2. **Substituting \( z \)**:
Substitute \( z \) into the equation:
\[ |r e^{i\theta} + \frac{2}{r e^{i\theta}}| = 2 \]
This simplifies to:
\[ |r e^{i\theta} + \frac{2}{r} e^{-i\theta}| = 2 \]

3. **Combine the Terms**:
We can express the left-hand side in terms of real and imaginary parts:
\[ |r \cos \theta + \frac{2}{r} \cos \theta + i \left( r \sin \theta - \frac{2}{r} \sin \theta \right)| = 2 \]
This can be simplified to:
\[ |(r + \frac{2}{r}) \cos \theta + i (r - \frac{2}{r}) \sin \theta| = 2 \]

4. **Using the Modulus**:
The modulus can be expressed as:
\[ \sqrt{(r + \frac{2}{r})^2 \cos^2 \theta + (r - \frac{2}{r})^2 \sin^2 \theta} = 2 \]

5. **Square Both Sides**:
Squaring both sides gives:
\[ (r + \frac{2}{r})^2 \cos^2 \theta + (r - \frac{2}{r})^2 \sin^2 \theta = 4 \]

6. **Expanding the Squares**:
Expanding both terms:
\[ (r^2 + 4 + \frac{4}{r^2}) \cos^2 \theta + (r^2 - 4 + \frac{4}{r^2}) \sin^2 \theta = 4 \]

7. **Combining Terms**:
Combine the terms:
\[ r^2 (\cos^2 \theta + \sin^2 \theta) + 4(\cos^2 \theta - \sin^2 \theta) + \frac{4}{r^2} (\cos^2 \theta + \sin^2 \theta) = 4 \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ r^2 + 4(\cos^2 \theta - \sin^2 \theta) + \frac{4}{r^2} = 4 \]

8. **Rearranging**:
Rearranging gives:
\[ r^2 + \frac{4}{r^2} + 4(\cos^2 \theta - \sin^2 \theta) = 4 \]

9. **Setting \( t = |z| = r \)**:
Let \( t = |z| \):
\[ t^2 + \frac{4}{t^2} + 4(\cos^2 \theta - \sin^2 \theta) = 4 \]

10. **Finding Maximum Value**:
To maximize \( t \), we analyze the quadratic:
\[ t^2 - 2t + 2 \leq 0 \]
The roots of this equation can be found using the quadratic formula:
\[ t = 1 \pm \sqrt{3} \]
Since \( t \) must be non-negative, we take:
\[ t \leq 1 + \sqrt{3} \]

### Conclusion:
Thus, the maximum value of \( |z| \) is:
\[ \boxed{1 + \sqrt{3}} \]