If the angle between the lines whose direction ratios are `(2,-1,2)` and `(x,3,5)` is `pi/4`, then the smaller value of x is

Options

A. 52
B. 4
C. 2
D. 1

Answer

To find the smaller value of \( x \) such that the angle between the lines with direction ratios \( (2, -1, 2) \) and \( (x, 3, 5) \) is \( \frac{\pi}{4} \), we can follow these steps:

### Step 1: Understand the angle between two lines
The angle \( \theta \) between two lines can be found using the formula:
\[ \cos \theta = \frac{n_1 \cdot n_2}{|n_1| |n_2|} \]
where \( n_1 \) and \( n_2 \) are the direction ratios of the two lines.

### Step 2: Set up the direction ratios
Let:
- \( n_1 = (2, -1, 2) \)
- \( n_2 = (x, 3, 5) \)

### Step 3: Calculate the dot product \( n_1 \cdot n_2 \)
The dot product is calculated as follows:
\[ n_1 \cdot n_2 = 2x + (-1) \cdot 3 + 2 \cdot 5 = 2x - 3 + 10 = 2x + 7 \]

### Step 4: Calculate the magnitudes \( |n_1| \) and \( |n_2| \)
The magnitude of \( n_1 \) is:
\[ |n_1| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

The magnitude of \( n_2 \) is:
\[ |n_2| = \sqrt{x^2 + 3^2 + 5^2} = \sqrt{x^2 + 9 + 25} = \sqrt{x^2 + 34} \]

### Step 5: Set up the equation using the angle
Given that \( \theta = \frac{\pi}{4} \), we have:
\[ \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
Thus,
\[ \frac{2x + 7}{3 \sqrt{x^2 + 34}} = \frac{1}{\sqrt{2}} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ (2x + 7) \sqrt{2} = 3 \sqrt{x^2 + 34} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (2x + 7)^2 \cdot 2 = 9(x^2 + 34) \]
Expanding both sides:
\[ 2(4x^2 + 28x + 49) = 9x^2 + 306 \]
\[ 8x^2 + 56x + 98 = 9x^2 + 306 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ 0 = 9x^2 - 8x^2 + 306 - 98 - 56x \]
\[ 0 = x^2 - 56x + 208 \]

### Step 9: Factor the quadratic equation
To factor \( x^2 - 56x + 208 = 0 \), we look for two numbers that multiply to \( 208 \) and add to \( -56 \). The factors are \( -52 \) and \( -4 \):
\[ (x - 52)(x - 4) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 52 = 0 \quad \Rightarrow \quad x = 52 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

### Conclusion
The smaller value of \( x \) is:
\[ \boxed{4} \]