The plane `x + 3y + 13 = 0 ` passes through the line of intersection of the planes `2x - 8y + 4z = p and 3x - 5y + 4z + 10 =0` . If the plane is perpendicular to the plane 3x - y - 2z - 4 = 0 , then the value of p is equal to

Options

A. 2
B. 5
C. 9
D. 3

Answer

To solve the problem, we need to find the value of \( p \) such that the plane \( x + 3y + 13 = 0 \) passes through the line of intersection of the planes \( 2x - 8y + 4z = p \) and \( 3x - 5y + 4z + 10 = 0 \), and is also perpendicular to the plane \( 3x - y - 2z - 4 = 0 \).

### Step-by-Step Solution:

1. **Identify the given planes**:
- Plane 1: \( 2x - 8y + 4z - p = 0 \)
- Plane 2: \( 3x - 5y + 4z + 10 = 0 \)
- Plane 3 (perpendicular): \( 3x - y - 2z - 4 = 0 \)

2. **Equation of the plane through the line of intersection**:
The equation of the plane that passes through the line of intersection of the two planes can be expressed as:
\[ 2x - 8y + 4z - p + \lambda(3x - 5y + 4z + 10) = 0 \]
Simplifying this gives:
\[ (2 + 3\lambda)x + (-8 - 5\lambda)y + (4 + 4\lambda)z + (-p + 10\lambda) = 0 \]

3. **Direction ratios of the planes**:
The direction ratios of the plane we derived are:
- Coefficients of \( x, y, z \): \( 2 + 3\lambda, -8 - 5\lambda, 4 + 4\lambda \)
- Direction ratios of Plane 3 are \( 3, -1, -2 \).

4. **Condition for perpendicularity**:
For the two planes to be perpendicular, the dot product of their direction ratios must equal zero:
\[ (2 + 3\lambda) \cdot 3 + (-8 - 5\lambda)(-1) + (4 + 4\lambda)(-2) = 0 \]

5. **Expanding the equation**:
Expanding the equation gives:
\[ 6 + 9\lambda + 8 + 5\lambda - 8 - 8\lambda = 0 \]
Simplifying this:
\[ 6 + 9\lambda + 8 + 5\lambda - 8 - 8\lambda = 0 \]
\[ 6 + 0\lambda = 0 \]
This simplifies to:
\[ 6 = 0 \quad \text{(which is incorrect)} \]

Correctly, we should have:
\[ 6 + 9\lambda + 8 + 5\lambda - 8 - 8\lambda = 0 \]
\[ 6 + 6\lambda = 0 \implies \lambda = -1 \]

6. **Substituting \( \lambda \) back**:
Substitute \( \lambda = -1 \) into the plane equation:
\[ (2 + 3(-1))x + (-8 - 5(-1))y + (4 + 4(-1))z + (-p + 10(-1)) = 0 \]
This simplifies to:
\[ -x - 3y + 0z - p - 10 = 0 \]
Rearranging gives:
\[ x + 3y + p + 10 = 0 \]

7. **Equate with the given plane**:
We know that this plane should equal \( x + 3y + 13 = 0 \):
\[ p + 10 = 13 \implies p = 3 \]

### Final Answer:
The value of \( p \) is \( \boxed{3} \).