Statement I Reflection of the point (5,1) in the line `x+y=0` is `(-1,-5)`
Statement II Reflection of a point `P(alpha,beta)` in the line `ax+by+c= 0 ` is ` Q (alpha',beta' ) " if " ((alpha +alpha')/2 ,(beta +beta' )/2)` lies on the line .

Options

A. Statement I is true ,statement II is true , statement II is a correct explanation for statement I
B. Statement I is true ,statement II is true statement II is not a correct explanation for statement I
C. Statement I is true ,statement II is false
D. Statement I is false ,statement II is true

Answer

To solve the problem, we need to verify both statements regarding the reflection of a point in a line.

### Step 1: Find the reflection of the point (5, 1) in the line x + y = 0.

1. **Identify the given point and line**:
- Point A = (5, 1)
- Line equation: x + y = 0

2. **Find the slope of the line**:
- The line x + y = 0 can be rewritten as y = -x. The slope (m) of this line is -1.

3. **Find the slope of the perpendicular line**:
- The slope of the line perpendicular to x + y = 0 is the negative reciprocal of -1, which is 1.

4. **Equation of the perpendicular line through point A (5, 1)**:
- Using the point-slope form:
\[ y - 1 = 1(x - 5) \implies y = x - 4 \]

5. **Find the intersection point (M) of the two lines**:
- Set the equations equal to each other:
\[ x - 4 = -x \implies 2x = 4 \implies x = 2 \]
- Substitute x = 2 into y = -x:
\[ y = -2 \]
- Thus, the intersection point M is (2, -2).

6. **Find the reflection point B (x1, y1)**:
- Since M is the midpoint of A and B, we have:
\[ M = \left(\frac{x_1 + 5}{2}, \frac{y_1 + 1}{2}\right) = (2, -2) \]
- This gives us two equations:
\[ \frac{x_1 + 5}{2} = 2 \implies x_1 + 5 = 4 \implies x_1 = -1 \]
\[ \frac{y_1 + 1}{2} = -2 \implies y_1 + 1 = -4 \implies y_1 = -5 \]
- Therefore, the reflection point B is (-1, -5).

### Step 2: Verify Statement I
- Statement I claims that the reflection of the point (5, 1) in the line x + y = 0 is (-1, -5).
- From our calculations, we found that the reflection point is indeed (-1, -5). Thus, Statement I is **correct**.

### Step 3: Verify Statement II
- Statement II states that the reflection of a point P(α, β) in the line ax + by + c = 0 is Q(α', β') if the midpoint ((α + α')/2, (β + β')/2) lies on the line.

1. **Using our earlier findings**:
- We found the midpoint M = (2, -2) for points A(5, 1) and B(-1, -5).
- We need to check if M lies on the line x + y = 0:
\[ 2 + (-2) = 0 \]
- Since this is true, it confirms that the midpoint lies on the line.

2. **Conclusion for Statement II**:
- Since we verified that the midpoint lies on the line, Statement II is also **correct**.

### Final Conclusion
Both Statement I and Statement II are correct.

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