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 1: Express the modulus condition
Given \(|z| = 2\), we can write:
\[
|z| = \sqrt{x^2 + y^2} = 2
\]
Squaring both sides, we get:
\[
x^2 + y^2 = 4
\]
### Step 2: Substitute \(z\) into the expression
We substitute \(z\) into the expression:
\[
\frac{z + 2}{z - 2} = \frac{(x + iy) + 2}{(x + iy) - 2} = \frac{(x + 2) + iy}{(x - 2) + iy}
\]
### Step 3: Rationalize the denominator
To find the imaginary part, we multiply the numerator and denominator by the conjugate of the denominator:
\[
\frac{((x + 2) + iy)((x - 2) - iy)}{((x - 2) + iy)((x - 2) - iy)}
\]
The denominator simplifies to:
\[
(x - 2)^2 + y^2 = (x - 2)^2 + (4 - x^2) = x^2 - 4x + 4 + 4 - x^2 = -4x + 8
\]
The numerator expands to:
\[
(x + 2)(x - 2) - i y (x + 2) + i y (x - 2) + y^2
\]
This simplifies to:
\[
x^2 - 4 - i(2y) + y^2 = (x^2 + y^2 - 4) - 2yi
\]
Using \(x^2 + y^2 = 4\), we have:
\[
0 - 2yi = -2yi
\]
### Step 4: Combine results
Thus, we have:
\[
\frac{-2yi}{-4x + 8} = \frac{2yi}{4x - 8}
\]
The imaginary part is:
\[
\frac{2y}{4x - 8}
\]
### Step 5: Analyze the conditions
Since \(xy \neq 0\), both \(x\) and \(y\) are non-zero. We need to find the values that this expression cannot take.
### Step 6: Determine the range of the imaginary part
The expression \(\frac{2y}{4x - 8}\) can take various values depending on the values of \(x\) and \(y\). However, we need to find values that are impossible given the conditions.
### Step 7: Check the options
The options given are 1, 2, 3, and 4. We analyze the expression:
- As \(y\) varies, the imaginary part can take various values.
- The critical points occur when \(4x - 8 = 0\) or \(x = 2\), which would make the expression undefined.
### Conclusion
After analyzing the expression and the conditions, we conclude that the imaginary part cannot be equal to 1, as it leads to contradictions based on the values of \(x\) and \(y\).
Thus, the final answer is:
\[
\text{The imaginary part of } \frac{z + 2}{z - 2} \text{ cannot be } 1.
\]
