To solve the problem, we need to find the values of the parameters \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) based on the given conditions, and then verify which statements are true.
### Step-by-step Solution:
1. **Understand the Given Conditions**:
- We have the condition: \(\alpha^2 + \beta^2 + \gamma^2 \neq 0\) and \(\alpha + \gamma = 1\).
- The points \(A(1, 0, -1)\) and \(C(3, 2, -1)\) are mirror images with respect to the plane defined by the equation \(\alpha x + \beta y + \gamma z = \delta\).
2. **Find the Midpoint**:
- The midpoint \(B\) between points \(A\) and \(C\) can be calculated as:
\[
B = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) = \left(\frac{1 + 3}{2}, \frac{0 + 2}{2}, \frac{-1 + (-1)}{2}\right) = (2, 1, -1)
\]
3. **Substitute Point B into the Plane Equation**:
- Since point \(B\) lies on the plane, we substitute its coordinates into the plane equation:
\[
\alpha(2) + \beta(1) + \gamma(-1) = \delta
\]
This simplifies to:
\[
2\alpha + \beta - \gamma = \delta \quad \text{(Equation 1)}
\]
4. **Find Direction Ratios**:
- The direction vector from \(A\) to \(C\) is given by:
\[
\text{Direction} = (3 - 1, 2 - 0, -1 - (-1)) = (2, 2, 0)
\]
- The normal vector of the plane is given by \((\alpha, \beta, \gamma)\). Thus, we can set up the following ratios:
\[
\frac{\alpha}{2} = \frac{\beta}{2} = \frac{\gamma}{0} = \lambda
\]
- From this, we can express:
\[
\alpha = 2\lambda, \quad \beta = 2\lambda, \quad \gamma = 0
\]
5. **Substituting into the Condition**:
- Using the condition \(\alpha + \gamma = 1\):
\[
2\lambda + 0 = 1 \implies \lambda = \frac{1}{2}
\]
- Therefore:
\[
\alpha = 2 \cdot \frac{1}{2} = 1, \quad \beta = 2 \cdot \frac{1}{2} = 1, \quad \gamma = 0
\]
6. **Finding Delta**:
- Substitute \(\alpha\), \(\beta\), and \(\gamma\) into Equation 1:
\[
2(1) + 1 - 0 = \delta \implies \delta = 3
\]
7. **Final Values**:
- We have found:
\[
\alpha = 1, \quad \beta = 1, \quad \gamma = 0, \quad \delta = 3
\]
8. **Check the Statements**:
- Now we check the provided statements:
1. \(\alpha + \beta = 2\) (True)
2. \(\delta - \gamma = 3\) (True)
3. \(\delta + \beta = 4\) (True)
4. \(\alpha + \beta + \gamma = 2\) (False)
### Conclusion:
The true statements are:
- \(\alpha + \beta = 2\)
- \(\delta - \gamma = 3\)
- \(\delta + \beta = 4\)
