The maximum value of |z| where z satisfies the condition` |z+(2/z)|=2` is

Options

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

Explanation

To find the maximum value of |z| given the condition |z + (2/z)| = 2, we can follow these steps:

### Step 1: Rewrite the given condition
We start with the equation:
\[ |z + \frac{2}{z}| = 2 \]

### Step 2: Let z = re^(iθ)
We can express z in polar form:
\[ z = re^{i\theta} \]
where \( r = |z| \) and \( \theta \) is the argument of z.

### Step 3: Substitute z into the equation
Substituting z into the equation gives:
\[ |re^{i\theta} + \frac{2}{re^{i\theta}}| = 2 \]
This simplifies to:
\[ |re^{i\theta} + \frac{2}{r} e^{-i\theta}| = 2 \]

### Step 4: Simplify the expression
Using the property of modulus:
\[ |re^{i\theta} + \frac{2}{r} e^{-i\theta}| = |r + \frac{2}{r} e^{-2i\theta}| \]
This can be expressed as:
\[ \sqrt{(r + \frac{2}{r} \cos(-2\theta))^2 + (\frac{2}{r} \sin(-2\theta))^2} = 2 \]

### Step 5: Square both sides
Squaring both sides gives:
\[ (r + \frac{2}{r} \cos(2\theta))^2 + (\frac{2}{r} \sin(2\theta))^2 = 4 \]

### Step 6: Expand and simplify
Expanding the left-hand side:
\[ r^2 + 2r \cdot \frac{2}{r} \cos(2\theta) + \frac{4}{r^2} \cos^2(2\theta) + \frac{4}{r^2} \sin^2(2\theta) = 4 \]
Using \(\cos^2(2\theta) + \sin^2(2\theta) = 1\):
\[ r^2 + 4 \cos(2\theta) + \frac{4}{r^2} = 4 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ r^2 + \frac{4}{r^2} + 4 \cos(2\theta) - 4 = 0 \]
This can be simplified to:
\[ r^2 + \frac{4}{r^2} + 4 \cos(2\theta) = 4 \]

### Step 8: Find the maximum value of |z|
To maximize |z|, we need to minimize \(4 \cos(2\theta)\). The minimum value of \(\cos(2\theta)\) is -1, hence:
\[ r^2 + \frac{4}{r^2} - 4 = 0 \]

### Step 9: Let \(x = r^2\)
Let \(x = r^2\), then we have:
\[ x + \frac{4}{x} - 4 = 0 \]
Multiplying through by \(x\) gives:
\[ x^2 - 4x + 4 = 0 \]
This factors to:
\[ (x - 2)^2 = 0 \]
Thus, \(x = 2\) or \(r^2 = 2\).

### Step 10: Calculate |z|
Taking the square root gives:
\[ r = |z| = \sqrt{2} \]

### Step 11: Maximum value of |z|
The maximum value of |z| is:
\[ \sqrt{2} + 1 \]

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