The planes `px+2y+2z-3=0 and 2x-y+z+2=0` intersect at an angle `pi/4`. What is the value of `p^2`?

Options

A. A) 24
B. B) 12
C. C) 6
D. D) 3

Answer

To find the value of \( p^2 \) given that the planes \( px + 2y + 2z - 3 = 0 \) and \( 2x - y + z + 2 = 0 \) intersect at an angle of \( \frac{\pi}{4} \), we can follow these steps:

### Step 1: Identify the coefficients of the planes
The first plane can be expressed as:
\[ A_1 = p, \quad B_1 = 2, \quad C_1 = 2 \]

The second plane can be expressed as:
\[ A_2 = 2, \quad B_2 = -1, \quad C_2 = 1 \]

### Step 2: Use the formula for the cosine of the angle between two planes
The cosine of the angle \( \theta \) between two planes is given by:
\[ \cos \theta = \frac{|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2} \sqrt{A_2^2 + B_2^2 + C_2^2}} \]
Given that \( \theta = \frac{\pi}{4} \), we have:
\[ \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]

### Step 3: Substitute the values into the formula
Substituting the coefficients into the formula:
\[ \frac{|p \cdot 2 + 2 \cdot (-1) + 2 \cdot 1|}{\sqrt{p^2 + 2^2 + 2^2} \sqrt{2^2 + (-1)^2 + 1^2}} = \frac{1}{\sqrt{2}} \]

### Step 4: Simplify the numerator
Calculating the numerator:
\[ |2p - 2 + 2| = |2p| \]

### Step 5: Simplify the denominators
Calculating the first denominator:
\[ \sqrt{p^2 + 4 + 4} = \sqrt{p^2 + 8} \]
Calculating the second denominator:
\[ \sqrt{4 + 1 + 1} = \sqrt{6} \]

### Step 6: Set up the equation
Now we can set up the equation:
\[ \frac{|2p|}{\sqrt{p^2 + 8} \cdot \sqrt{6}} = \frac{1}{\sqrt{2}} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ |2p| \cdot \sqrt{2} = \sqrt{6} \cdot \sqrt{p^2 + 8} \]

### Step 8: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ (2p)^2 \cdot 2 = 6(p^2 + 8) \]
This simplifies to:
\[ 8p^2 = 6p^2 + 48 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 8p^2 - 6p^2 = 48 \]
\[ 2p^2 = 48 \]

### Step 10: Solve for \( p^2 \)
Dividing both sides by 2:
\[ p^2 = 24 \]

### Final Answer
Thus, the value of \( p^2 \) is:
\[ \boxed{24} \]