If the direction ratio of two lines are given by `3lm-4ln+mn=0 and l+2m+3n=0`, then the angle between the lines, is

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(2)`

Answer

To find the angle between the two lines given by their direction ratios, we will follow these steps:

### Step 1: Write down the equations of the lines
The equations of the lines are given as:
1. \( 3lm - 4ln + mn = 0 \) (Equation 1)
2. \( l + 2m + 3n = 0 \) (Equation 2)

### Step 2: Solve for one variable
From Equation 2, we can express \( l \) in terms of \( m \) and \( n \):
\[ l = -2m - 3n \]

### Step 3: Substitute \( l \) into Equation 1
Now, substitute \( l \) into Equation 1:
\[ 3(-2m - 3n)m - 4(-2m - 3n)n + mn = 0 \]
This simplifies to:
\[ -6m^2 - 9mn + 8mn + 12n^2 + mn = 0 \]
Combining like terms gives:
\[ -6m^2 + 0mn + 12n^2 = 0 \]

### Step 4: Factor the equation
We can factor out a common term:
\[ -6m^2 + 12n^2 = 0 \]
Dividing through by 6 gives:
\[ - m^2 + 2n^2 = 0 \]
Thus:
\[ m^2 = 2n^2 \]

### Step 5: Find the relationship between \( m \) and \( n \)
Taking the square root gives:
\[ m = \sqrt{2}n \quad \text{or} \quad m = -\sqrt{2}n \]

### Step 6: Find the direction ratios
Substituting \( m = \sqrt{2}n \) into the expression for \( l \):
\[ l = -2(\sqrt{2}n) - 3n = -2\sqrt{2}n - 3n = (-2\sqrt{2} - 3)n \]
Thus, the direction ratios of the first line can be represented as:
\[ (-2\sqrt{2} - 3, \sqrt{2}, 1) \]

For \( m = -\sqrt{2}n \):
\[ l = -2(-\sqrt{2}n) - 3n = 2\sqrt{2}n - 3n = (2\sqrt{2} - 3)n \]
Thus, the direction ratios of the second line can be represented as:
\[ (2\sqrt{2} - 3, -\sqrt{2}, 1) \]

### Step 7: Calculate the angle between the lines
The angle \( \theta \) between two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) can be found using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

Substituting the direction ratios:
1. For the first line: \( a_1 = -2\sqrt{2} - 3, b_1 = \sqrt{2}, c_1 = 1 \)
2. For the second line: \( a_2 = 2\sqrt{2} - 3, b_2 = -\sqrt{2}, c_2 = 1 \)

Calculating the dot product:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = (-2\sqrt{2} - 3)(2\sqrt{2} - 3) + (\sqrt{2})(-\sqrt{2}) + (1)(1) \]

### Step 8: Simplify the expression
Calculating the dot product step by step:
1. \( (-2\sqrt{2})(2\sqrt{2}) = -8 \)
2. \( (-2\sqrt{2})(-3) = 6\sqrt{2} \)
3. \( (-3)(2\sqrt{2}) = -6\sqrt{2} \)
4. \( (-3)(-3) = 9 \)
5. \( -2 + 1 = -1 \)

Combining gives:
\[ -8 + 9 + 0 = 1 \]

### Step 9: Calculate magnitudes
Calculating the magnitudes:
\[ \sqrt{(-2\sqrt{2} - 3)^2 + (\sqrt{2})^2 + 1^2} \quad \text{and} \quad \sqrt{(2\sqrt{2} - 3)^2 + (-\sqrt{2})^2 + 1^2} \]

### Final Step: Calculate \( \theta \)
After substituting and simplifying, we find that:
\[ \cos \theta = 0 \]
This implies:
\[ \theta = \frac{\pi}{2} \]

### Conclusion
The angle between the two lines is \( \frac{\pi}{2} \) radians.

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