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 \( |z| \) for the complex number \( z \) that satisfies the condition \( |z + \frac{2}{z}| = 2 \).

### Step 1: Rewrite the Given Condition
We start with the equation:
\[ |z + \frac{2}{z}| = 2 \]
Let \( z = re^{i\theta} \), where \( r = |z| \) and \( \theta \) is the argument of \( z \). Then we can rewrite \( \frac{2}{z} \) as:
\[ \frac{2}{z} = \frac{2}{re^{i\theta}} = \frac{2}{r} e^{-i\theta} \]
Thus, the expression becomes:
\[ |re^{i\theta} + \frac{2}{r} e^{-i\theta}| = 2 \]

### Step 2: Combine the Terms
We can express this as:
\[ |r e^{i\theta} + \frac{2}{r} e^{-i\theta}| = |r \cos \theta + \frac{2}{r} \cos(-\theta) + i(r \sin \theta + \frac{2}{r} \sin(-\theta))| \]
This simplifies to:
\[ |r \cos \theta + \frac{2}{r} \cos \theta + i(r \sin \theta - \frac{2}{r} \sin \theta)| = |(r + \frac{2}{r}) \cos \theta + i(r - \frac{2}{r}) \sin \theta| \]

### Step 3: Apply the Modulus
The modulus can be calculated as:
\[ \sqrt{((r + \frac{2}{r}) \cos \theta)^2 + ((r - \frac{2}{r}) \sin \theta)^2} = 2 \]
Squaring both sides gives:
\[ (r + \frac{2}{r})^2 \cos^2 \theta + (r - \frac{2}{r})^2 \sin^2 \theta = 4 \]

### Step 4: Expand and Simplify
Expanding both terms:
\[ (r^2 + 4 + \frac{4}{r^2}) \cos^2 \theta + (r^2 - 4 + \frac{4}{r^2}) \sin^2 \theta = 4 \]
Combining terms yields:
\[ r^2(\cos^2 \theta + \sin^2 \theta) + 4 \cos^2 \theta - 4 \sin^2 \theta + \frac{4}{r^2}(\cos^2 \theta + \sin^2 \theta) = 4 \]
Using \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ r^2 + 4 \cos^2 \theta - 4 \sin^2 \theta + \frac{4}{r^2} = 4 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ r^2 + \frac{4}{r^2} + 4(\cos^2 \theta - \sin^2 \theta) = 4 \]
Let \( x = r^2 \):
\[ x + \frac{4}{x} + 4(\cos^2 \theta - \sin^2 \theta) = 4 \]
This implies:
\[ x + \frac{4}{x} = 4 - 4(\cos^2 \theta - \sin^2 \theta) \]

### Step 6: Finding Maximum Value
To maximize \( |z| \), we need to minimize \( \cos^2 \theta - \sin^2 \theta \). The minimum value occurs when \( \theta = \frac{\pi}{4} \) or \( \theta = \frac{3\pi}{4} \), giving \( \cos^2 \theta - \sin^2 \theta = 0 \).

Thus, we have:
\[ x + \frac{4}{x} = 4 \]
Multiplying by \( x \):
\[ x^2 - 4x + 4 = 0 \]
This factors to:
\[ (x - 2)^2 = 0 \]
Thus, \( x = 2 \) which means \( r^2 = 2 \) or \( |z| = \sqrt{2} \).

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