If Re `(lambda Z + (2)/(Z))` is zero and Re(z) `!= 0` and |Z| = 2. Then `lambda` can be

Options

A. 2
B. `-2`
C. `(1)/(2)`
D. `-(1)/(2)`

Explanation

To solve the problem, we need to find the value of \(\lambda\) given that the real part of the expression \(\lambda z + \frac{2}{z}\) is zero, where \(z\) is a complex number with \(|z| = 2\) and \(\text{Re}(z) \neq 0\).

### Step-by-Step Solution

1. **Express \(z\) in terms of its components:**
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers.

2. **Calculate \(|z|\):**
Given \(|z| = 2\), we have:
\[ |z| = \sqrt{x^2 + y^2} = 2 \implies x^2 + y^2 = 4. \]

3. **Substitute \(z\) into the expression:**
We need to evaluate the expression \(\lambda z + \frac{2}{z}\):
\[ \lambda z + \frac{2}{z} = \lambda(x + iy) + \frac{2}{x + iy}. \]

4. **Rationalize \(\frac{2}{z}\):**
To simplify \(\frac{2}{z}\), we multiply by the conjugate:
\[ \frac{2}{z} = \frac{2}{x + iy} \cdot \frac{x - iy}{x - iy} = \frac{2(x - iy)}{x^2 + y^2} = \frac{2(x - iy)}{4} = \frac{x}{2} - \frac{iy}{2}. \]

5. **Combine the terms:**
Now substitute back into the expression:
\[ \lambda(x + iy) + \left(\frac{x}{2} - \frac{iy}{2}\right) = \lambda x + \frac{x}{2} + i(\lambda y - \frac{y}{2}). \]

6. **Separate the real and imaginary parts:**
The real part is:
\[ \text{Re}(\lambda z + \frac{2}{z}) = \lambda x + \frac{x}{2}. \]
The imaginary part is:
\[ \text{Im}(\lambda z + \frac{2}{z}) = \lambda y - \frac{y}{2}. \]

7. **Set the real part to zero:**
According to the problem, we set the real part to zero:
\[ \lambda x + \frac{x}{2} = 0. \]

8. **Factor out \(x\):**
Since \(\text{Re}(z) \neq 0\), we can divide by \(x\):
\[ \lambda + \frac{1}{2} = 0 \implies \lambda = -\frac{1}{2}. \]

9. **Conclusion:**
The value of \(\lambda\) is:
\[ \lambda = -\frac{1}{2}. \]