Let z be a complex number such that |z| = 2, then maximum possible value of `|z + (2)/(z)|` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

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

### Step-by-step Solution:

1. **Understanding the Given Information**:
We know that \( z \) is a complex number such that \( |z| = 2 \). This means that \( z \) lies on a circle of radius 2 in the complex plane.

2. **Rewriting the Expression**:
We want to maximize the expression \( |z + \frac{2}{z}| \). We can rewrite \( \frac{2}{z} \) using the property of modulus:
\[ \left| \frac{2}{z} \right| = \frac{2}{|z|} = \frac{2}{2} = 1 \]
Therefore, we can express the original expression as:
\[ |z + \frac{2}{z}| = |z + w| \]
where \( w = \frac{2}{z} \) and \( |w| = 1 \).

3. **Using the Triangle Inequality**:
By the triangle inequality, we have:
\[ |z + w| \leq |z| + |w| \]
Substituting the values we know:
\[ |z + w| \leq |z| + |w| = 2 + 1 = 3 \]

4. **Finding the Maximum Value**:
To check if the maximum value of 3 can actually be achieved, we need to find \( z \) and \( w \) such that they are in the same direction. This occurs when:
\[ z = 2e^{i\theta} \quad \text{and} \quad w = e^{i\theta} \]
for some angle \( \theta \). Thus:
\[ |z + w| = |2e^{i\theta} + e^{i\theta}| = |(2 + 1)e^{i\theta}| = |3e^{i\theta}| = 3 \]
Therefore, the maximum value of \( |z + \frac{2}{z}| \) is indeed 3.

5. **Conclusion**:
Hence, the maximum possible value of \( |z + \frac{2}{z}| \) is \( \boxed{3} \).