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 step by step, we will follow the reasoning provided in the video transcript and derive the value of \( p \).

### Step 1: Understand the planes involved
We have the following planes:
1. Plane 1: \( 2x - 8y + 4z = p \)
2. Plane 2: \( 3x - 5y + 4z + 10 = 0 \)
3. Plane 3: \( 3x - y - 2z - 4 = 0 \)

### Step 2: Form the equation of the required plane
The required plane passes through the line of intersection of Plane 1 and Plane 2. The equation of the required plane can be expressed as:
\[ 2x - 8y + 4z - p + \lambda (3x - 5y + 4z + 10) = 0 \]
where \( \lambda \) is a parameter.

### Step 3: Combine the coefficients
Combining the coefficients of \( x \), \( y \), \( z \), and the constant term, we get:
\[ (2 + 3\lambda)x + (-8 - 5\lambda)y + (4 + 4\lambda)z + (10\lambda - p) = 0 \]

### Step 4: Identify the direction ratios
The direction ratios of the required plane are:
- For \( x \): \( 2 + 3\lambda \)
- For \( y \): \( -8 - 5\lambda \)
- For \( z \): \( 4 + 4\lambda \)

The direction ratios of Plane 3 are:
- \( 3, -1, -2 \)

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

### Step 6: Expand and simplify the equation
Expanding the equation gives:
\[ 3(2 + 3\lambda) + (8 + 5\lambda) - 2(4 + 4\lambda) = 0 \]
This simplifies to:
\[ 6 + 9\lambda + 8 + 5\lambda - 8 - 8\lambda = 0 \]
Combining like terms results in:
\[ 6\lambda + 6 = 0 \]
Thus, we find:
\[ 6\lambda = -6 \implies \lambda = -1 \]

### Step 7: Substitute \( \lambda \) back into the plane equation
Now substitute \( \lambda = -1 \) back into the equation of the required plane:
\[ (2 + 3(-1))x + (-8 - 5(-1))y + (4 + 4(-1))z + (10(-1) - p) = 0 \]
This simplifies to:
\[ (-1)x + (-3)y + 0z + (-10 - p) = 0 \]
or:
\[ x + 3y + (10 + p) = 0 \]

### Step 8: Set the equation equal to the given plane
We know from the problem statement that this plane is equivalent to:
\[ x + 3y + 13 = 0 \]
Thus, we can equate:
\[ 10 + p = 13 \]

### Step 9: Solve for \( p \)
Solving for \( p \) gives:
\[ p = 13 - 10 = 3 \]

### Final Answer
The value of \( p \) is \( 3 \).