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 the conditions on the complex number \( z \).

### Step-by-Step Solution:

1. **Express \( z \) in terms of its real and imaginary parts**:
Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \).

2. **Use the modulus condition**:
We know that \( |z| = 2 \). Therefore, we have:
\[ |z| = \sqrt{x^2 + y^2} = 2 \]
Squaring both sides gives:
\[ x^2 + y^2 = 4 \]

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

4. **Simplify \( \frac{2}{z} \)**:
To simplify \( \frac{2}{z} \), multiply the numerator and denominator by the conjugate:
\[ \frac{2}{x + iy} = \frac{2(x - iy)}{x^2 + y^2} = \frac{2x}{x^2 + y^2} - \frac{2iy}{x^2 + y^2} \]
Since \( x^2 + y^2 = 4 \), we can substitute this into the equation:
\[ \frac{2}{z} = \frac{2x}{4} - \frac{2iy}{4} = \frac{x}{2} - \frac{iy}{2} \]

5. **Combine the terms**:
Now, we can combine \( \lambda z \) and \( \frac{2}{z} \):
\[ \lambda z + \frac{2}{z} = \lambda (x + iy) + \left( \frac{x}{2} - \frac{iy}{2} \right) \]
This gives:
\[ = \lambda x + \frac{x}{2} + i \left( \lambda y - \frac{y}{2} \right) \]

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

7. **Set the real part to zero**:
We are given that the real part is zero:
\[ \lambda x + \frac{x}{2} = 0 \]
Factor out \( x \) (noting that \( x \neq 0 \)):
\[ x \left( \lambda + \frac{1}{2} \right) = 0 \]
Thus, we have:
\[ \lambda + \frac{1}{2} = 0 \implies \lambda = -\frac{1}{2} \]

8. **Conclusion**:
The value of \( \lambda \) that satisfies the given conditions is:
\[ \lambda = -\frac{1}{2} \]