The plane denoted by `P_1 : 4x+7y+4z+81=0` is rotated through a right angle about its line of intersection with plane `P_2 : 5x+3y+10z=25`. If the plane in its new position be denoted by P, and the distance of this plane from the origin is d, then the value of `[(k)/(2)]` (where[.] represents greatest integer less than or equal to k) is....

Answer

To solve the problem, we need to follow these steps:

### Step 1: Identify the equations of the planes
The equations of the two planes are given as:
- Plane \( P_1: 4x + 7y + 4z + 81 = 0 \)
- Plane \( P_2: 5x + 3y + 10z - 25 = 0 \)

### Step 2: Find the line of intersection of the two planes
The line of intersection of two planes can be represented using a parameter \( \lambda \):
\[ P = P_1 + \lambda P_2 \]
Substituting the equations of the planes into this expression gives:
\[ P = (4x + 7y + 4z + 81) + \lambda (5x + 3y + 10z - 25) = 0 \]

### Step 3: Combine the equations
Combining the equations, we have:
\[ (4 + 5\lambda)x + (7 + 3\lambda)y + (4 + 10\lambda)z + (81 - 25\lambda) = 0 \]

### Step 4: Determine the condition for perpendicularity
For the new plane \( P \) to be perpendicular to \( P_1 \), the dot product of their normal vectors must be zero. The normal vector of \( P_1 \) is \( (4, 7, 4) \) and the normal vector of \( P \) is \( (4 + 5\lambda, 7 + 3\lambda, 4 + 10\lambda) \).

Setting the dot product to zero:
\[ 4(4 + 5\lambda) + 7(7 + 3\lambda) + 4(4 + 10\lambda) = 0 \]

### Step 5: Simplify the equation
Expanding this gives:
\[ 16 + 20\lambda + 49 + 21\lambda + 16 + 40\lambda = 0 \]
Combining like terms:
\[ (20 + 21 + 40)\lambda + (16 + 49 + 16) = 0 \]
\[ 81\lambda + 81 = 0 \]
Thus, we find:
\[ \lambda = -1 \]

### Step 6: Substitute \( \lambda \) back into the plane equation
Substituting \( \lambda = -1 \) into the equation for \( P \):
\[ P = (4 - 5)x + (7 - 3)y + (4 - 10)z + (81 + 25) = 0 \]
This simplifies to:
\[ -x + 4y - 6z + 106 = 0 \]

### Step 7: Find the distance from the origin to the new plane
The distance \( d \) from the origin to the plane \( Ax + By + Cz + D = 0 \) is given by:
\[ d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} \]
Here, \( A = -1, B = 4, C = -6, D = 106 \):
\[ d = \frac{|106|}{\sqrt{(-1)^2 + 4^2 + (-6)^2}} = \frac{106}{\sqrt{1 + 16 + 36}} = \frac{106}{\sqrt{53}} \]

### Step 8: Calculate \( \frac{d}{2} \)
Now, we need to find \( \frac{d}{2} \):
\[ \frac{d}{2} = \frac{106}{2\sqrt{53}} = \frac{53}{\sqrt{53}} = \sqrt{53} \]

### Step 9: Find the greatest integer less than or equal to \( k \)
The greatest integer less than or equal to \( \sqrt{53} \) can be calculated. Since \( \sqrt{53} \) is approximately \( 7.28 \), we have:
\[ \lfloor \sqrt{53} \rfloor = 7 \]

### Final Answer
Thus, the value of \( \left\lfloor \frac{k}{2} \right\rfloor \) is:
\[ \boxed{7} \]