The plane `4x+7y+4z+81=0` is rotated through a right angle about its line of intersection with the plane `5x+3y+10 z=25.` The equation of the plane in its new position is `x-4y+6z=k` where `k` is

Options

A. `x-4y+6z=106`
B. `x-8y+13z=103`
C. `x-4y+6z=110`
D. `x-8y+13z=19=105`

Explanation

To solve the problem, we need to find the value of \( k \) in the equation of the new plane after the original plane has been rotated through a right angle about the line of intersection with another plane.

### Step-by-Step Solution:

1. **Identify the given planes**:
- The first plane is given by the equation:
\[ 4x + 7y + 4z + 81 = 0 \]
- The second plane is given by the equation:
\[ 5x + 3y + 10z = 25 \]

2. **Find the normal vectors of the planes**:
- The normal vector of the first plane \( (4x + 7y + 4z + 81 = 0) \) is:
\[ \mathbf{n_1} = \langle 4, 7, 4 \rangle \]
- The normal vector of the second plane \( (5x + 3y + 10z - 25 = 0) \) is:
\[ \mathbf{n_2} = \langle 5, 3, 10 \rangle \]

3. **Find the line of intersection of the two planes**:
- The line of intersection can be represented as a linear combination of the normal vectors. The equation of the new plane after rotation can be expressed as:
\[ 4x + 7y + 4z + 81 + \lambda (5x + 3y + 10z - 25) = 0 \]

4. **Combine the equations**:
- Expanding the equation gives:
\[ (4 + 5\lambda)x + (7 + 3\lambda)y + (4 + 10\lambda)z + (81 - 25\lambda) = 0 \]

5. **Condition for perpendicularity**:
- Since the new plane is perpendicular to the original plane, the dot product of their normal vectors must equal zero:
\[ (4 + 5\lambda) \cdot 4 + (7 + 3\lambda) \cdot 7 + (4 + 10\lambda) \cdot 4 = 0 \]

6. **Set up the equation**:
- Expanding this gives:
\[ 16 + 20\lambda + 49 + 21\lambda + 16 + 40\lambda = 0 \]
- Combine like terms:
\[ 81 + 81\lambda = 0 \]

7. **Solve for \( \lambda \)**:
- Rearranging gives:
\[ 81\lambda = -81 \implies \lambda = -1 \]

8. **Substitute \( \lambda \) back into the plane equation**:
- Substitute \( \lambda = -1 \) into the combined equation:
\[ (4 + 5(-1))x + (7 + 3(-1))y + (4 + 10(-1))z + (81 - 25(-1)) = 0 \]
- This simplifies to:
\[ -x + 4y - 6z + 106 = 0 \]

9. **Rearranging gives the new plane equation**:
- The equation of the new plane can be written as:
\[ x - 4y + 6z = 106 \]

10. **Identify \( k \)**:
- From the equation \( x - 4y + 6z = k \), we find that:
\[ k = 106 \]

### Final Answer:
The value of \( k \) is \( 106 \).