Statement-I The point `A(3, 1, 6)` is the mirror image of the point `B(1, 3, 4)` in the plane `x-y+z=5`.
Statement-II The plane `x-y+z=5` bisect the line segment joining `A(3, 1, 6) and B(1,3, 4)`.

Options

A. Statement-I is true, Statement II is also true, Statement-II is the correct explanation of Statement-I.
B. Statement-I is true, Statement-II is also true, Statement-II is not the correct explanation of 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 analyze both statements regarding the points A(3, 1, 6) and B(1, 3, 4) in relation to the plane defined by the equation x - y + z = 5.

### Step 1: Find the Midpoint of Line Segment AB
The midpoint M of the line segment joining points A and B can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Where \( A(3, 1, 6) \) and \( B(1, 3, 4) \).

Calculating the coordinates:
\[ M = \left( \frac{3 + 1}{2}, \frac{1 + 3}{2}, \frac{6 + 4}{2} \right) = \left( \frac{4}{2}, \frac{4}{2}, \frac{10}{2} \right) = (2, 2, 5) \]

### Step 2: Check if the Midpoint Lies on the Plane
We need to check if the midpoint M(2, 2, 5) lies on the plane defined by the equation \( x - y + z = 5 \).
Substituting the coordinates of M into the plane equation:
\[ 2 - 2 + 5 = 5 \]
This simplifies to:
\[ 5 = 5 \]
Thus, the midpoint M lies on the plane.

### Step 3: Find the Direction Ratios of Line AB
The direction ratios of the line segment AB can be calculated as follows:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (1 - 3, 3 - 1, 4 - 6) = (-2, 2, -2) \]

### Step 4: Find the Normal Vector of the Plane
The normal vector of the plane \( x - y + z = 5 \) can be derived from the coefficients of x, y, and z in the equation:
\[ \text{Normal vector} = (1, -1, 1) \]

### Step 5: Check if Line AB is Perpendicular to the Plane
To check if line AB is perpendicular to the plane, we can take the dot product of the direction ratios of AB and the normal vector of the plane:
\[ \text{Dot product} = (-2)(1) + (2)(-1) + (-2)(1) = -2 - 2 - 2 = -6 \]
Since the dot product is not zero, line AB is not perpendicular to the plane.

### Conclusion
- **Statement I**: The point A(3, 1, 6) is the mirror image of point B(1, 3, 4) in the plane \( x - y + z = 5 \) is **True** because the midpoint lies on the plane.
- **Statement II**: The plane \( x - y + z = 5 \) bisects the line segment joining A and B is also **True** as the midpoint lies on the plane.

However, Statement II is not the correct explanation for Statement I because the perpendicularity condition is not satisfied.

### Final Answer
Both statements are true, but Statement II is not the correct explanation for Statement I.