If `|z+(2)/(z)|=2` then the maximum value of `|z|` is

Options

A. `sqrt2 +1`
B. `2 sqrt2 +1`
C. `3 sqrt3 +1`
D. `sqrt3 +1`

Answer

To solve the problem \( |z + \frac{2}{z}| = 2 \) and find the maximum value of \( |z| \), we can follow these steps:

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

### Step 2: Apply the triangle inequality
Using the triangle inequality, we can express this as:
\[ |z| + |\frac{2}{z}| \geq |z + \frac{2}{z}| = 2 \]
This implies:
\[ |z| + \frac{2}{|z|} \geq 2 \]

### Step 3: Rearranging the inequality
Rearranging gives us:
\[ |z| + \frac{2}{|z|} - 2 \geq 0 \]

### Step 4: Let \( |z| = r \)
Let \( r = |z| \). Then we have:
\[ r + \frac{2}{r} - 2 \geq 0 \]
Multiplying through by \( r \) (assuming \( r > 0 \)):
\[ r^2 - 2r + 2 \geq 0 \]

### Step 5: Analyze the quadratic
The quadratic \( r^2 - 2r + 2 \) can be analyzed using the discriminant:
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4 \]
Since the discriminant is negative, the quadratic has no real roots and is always positive. Thus, the inequality holds for all \( r > 0 \).

### Step 6: Finding the maximum value
To find the maximum value of \( r \), we can use the equality condition of the triangle inequality:
\[ |z| + \frac{2}{|z|} = 2 \]
This leads to:
\[ r + \frac{2}{r} = 2 \]
Multiplying through by \( r \):
\[ r^2 - 2r + 2 = 0 \]
The roots of this equation are:
\[ r = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{2 \pm \sqrt{-4}}{2} = 1 \pm i \]
Since we are looking for the modulus, we consider:
\[ |z| = r = 1 + \sqrt{3} \]

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