If `z=x+iy, AA x,y in R, i^(2)=-1, xy ne 0 and |z|=2`, then the imaginary part of `(z+2)/(z-2)` cannot be

Options

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

Answer

To solve the problem, we need to find the imaginary part of the expression \((z + 2)/(z - 2)\) given that \(z = x + iy\) where \(x, y \in \mathbb{R}\), \(xy \neq 0\), and \(|z| = 2\).

### Step-by-Step Solution:

1. **Understanding the Modulus Condition**:
Given \(|z| = 2\), we know:
\[ |z| = \sqrt{x^2 + y^2} = 2 \]
Squaring both sides gives:
\[ x^2 + y^2 = 4 \]

2. **Substituting \(z\)**:
We can express \(z + 2\) and \(z - 2\):
\[ z + 2 = (x + 2) + iy \]
\[ z - 2 = (x - 2) + iy \]

3. **Forming the Expression**:
Now we can write the expression:
\[ \frac{z + 2}{z - 2} = \frac{(x + 2) + iy}{(x - 2) + iy} \]

4. **Multiplying by the Conjugate**:
To simplify, multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{((x + 2) + iy)((x - 2) - iy)}{((x - 2) + iy)((x - 2) - iy)} \]

5. **Calculating the Denominator**:
The denominator simplifies as follows:
\[ (x - 2)^2 + y^2 = x^2 - 4x + 4 + y^2 = (x^2 + y^2) - 4x + 4 = 4 - 4x + 4 = 8 - 4x \]

6. **Calculating the Numerator**:
The numerator expands to:
\[ (x + 2)(x - 2) - i(y(x - 2) + y(x + 2)) = (x^2 - 4) + i(y(x - 2) - y(x + 2)) \]
Simplifying the imaginary part:
\[ y(x - 2) - y(x + 2) = -4y \]
Thus, the numerator becomes:
\[ (x^2 - 4) - 4yi \]

7. **Combining the Results**:
Now we have:
\[ \frac{(x^2 - 4) - 4yi}{8 - 4x} \]
This can be separated into real and imaginary parts:
\[ \text{Real part: } \frac{x^2 - 4}{8 - 4x}, \quad \text{Imaginary part: } \frac{-4y}{8 - 4x} \]

8. **Finding the Imaginary Part**:
The imaginary part is:
\[ \text{Imaginary part} = \frac{-4y}{8 - 4x} \]

9. **Analyzing the Conditions**:
Given \(x^2 + y^2 = 4\) and \(xy \neq 0\), we can analyze the values of \(y\) and \(x\). Since \(y\) cannot be zero, we can conclude that \(y\) must be non-zero.

10. **Determining the Range of the Imaginary Part**:
Since \(x\) can take values in the range determined by \(x^2 + y^2 = 4\) and \(xy \neq 0\), we can find that the imaginary part cannot equal certain values based on the conditions provided.

### Conclusion:
After analyzing the conditions, we conclude that the imaginary part of \((z + 2)/(z - 2)\) cannot be equal to \(1\) or \(-1\). Thus, the answer is:

**The imaginary part of \((z + 2)/(z - 2)\) cannot be \(1\) or \(-1\).**