Statement-1: Reflection of the point (-3, 2) in the line x + y = 0 is (-2, 3).Statement-2: The reflection of a point `P(alpha,beta)` in the line ax+by+c=0 is the point `Q(alpha', beta')`if `((alpha+alpha')/2, (beta+beta')/2)` lies on the line

Options

A. Statement -1 is True , Statement - 2 is true , Statement- 2 is a correct explanation for statement - 2
B. Statement-1 is True , Statement-2 is True , Statement -2 is not a correct explanation for Statement - 1 .
C. Statement-1 is True , Statement - 2 is False .
D. Statement - 1 is False , Statement -2 is True .

Explanation

To solve the problem, we need to analyze the two statements provided and verify their correctness step by step.

### Step 1: Understanding Statement 2
Statement 2 states that the reflection of a point \( P(\alpha, \beta) \) in the line \( ax + by + c = 0 \) results in the point \( Q(\alpha', \beta') \) if the midpoint \( M \left( \frac{\alpha + \alpha'}{2}, \frac{\beta + \beta'}{2} \right) \) lies on the line.

**Hint:** Remember that the midpoint of a segment is the average of the coordinates of the endpoints.

### Step 2: Finding the Midpoint
Let \( P(\alpha, \beta) \) be the point and \( Q(\alpha', \beta') \) be its reflection. The midpoint \( M \) is given by:
\[ M = \left( \frac{\alpha + \alpha'}{2}, \frac{\beta + \beta'}{2} \right) \]
According to the statement, this midpoint must satisfy the line equation \( ax + by + c = 0 \).

**Hint:** Substitute the coordinates of midpoint \( M \) into the line equation to check if it lies on the line.

### Step 3: Verifying Statement 1
Statement 1 claims that the reflection of the point \( (-3, 2) \) in the line \( x + y = 0 \) is \( (-2, 3) \).

1. **Identify the line:** The line \( x + y = 0 \) can be rewritten as \( y = -x \).
2. **Find the slope:** The slope of the line is \( -1 \).
3. **Find the slope of the line connecting \( P \) and \( Q \):** Since the reflection line is the mirror, the line connecting \( P \) and \( Q \) must be perpendicular to it. Therefore, the slope of line \( PQ \) should be \( 1 \) (the negative reciprocal of \( -1 \)).

**Hint:** Use the point-slope form to find the equation of the line through \( P(-3, 2) \) with slope \( 1 \).

### Step 4: Finding the Intersection Point
To find the intersection point \( M \) of the line \( PQ \) and the line \( x + y = 0 \):
1. **Equation of line \( PQ \):**
\[ y - 2 = 1(x + 3) \implies y = x + 5 \]
2. **Set the equations equal to find \( M \):**
\[ x + 5 = -x \implies 2x = -5 \implies x = -\frac{5}{2} \]
\[ y = -\left(-\frac{5}{2}\right) = \frac{5}{2} \]
Thus, \( M = \left(-\frac{5}{2}, \frac{5}{2}\right) \).

**Hint:** Ensure that both lines intersect at the correct point by substituting back into both equations.

### Step 5: Using the Midpoint Formula
Now, using the midpoint formula:
\[ M = \left( \frac{-3 + \alpha'}{2}, \frac{2 + \beta'}{2} \right) \]
Set this equal to \( M \):
\[ \frac{-3 + \alpha'}{2} = -\frac{5}{2} \quad \text{and} \quad \frac{2 + \beta'}{2} = \frac{5}{2} \]
From these equations, solve for \( \alpha' \) and \( \beta' \):
1. From the first equation:
\[ -3 + \alpha' = -5 \implies \alpha' = -2 \]
2. From the second equation:
\[ 2 + \beta' = 5 \implies \beta' = 3 \]

**Hint:** Check your calculations to ensure that the values obtained for \( \alpha' \) and \( \beta' \) match the reflection point given in Statement 1.

### Conclusion
Since we have verified that the reflection of point \( (-3, 2) \) in the line \( x + y = 0 \) is indeed \( (-2, 3) \), both statements are true. Therefore, Statement 2 correctly explains Statement 1.

**Final Answer:** Both Statement 1 and Statement 2 are true, and Statement 2 is the correct explanation for Statement 1.