Question

If the direction cosines of two lines satisfy the equations \( 2l+m-n=0 \), \( l^2-2m^2+n^2=0 \) and \( \theta \) is the angle between the lines then \( \cos\theta = \)

• A. \( \frac{1}{5} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{\pi}{3} \)

Hint:

Factorizing the homogeneous quadratic equation is the standard method for finding the two sets of direction ratios.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

We solve the linear and quadratic equations to find the direction ratios of the two lines. The cosine of the angle is the dot product of the normalized direction vectors.
Step 2: Key Formula or Approach:

1. Substitution to find ratios. 2. \( \cos \theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}} \).
Step 3: Detailed Explanation:

Given \( n = 2l + m \). Substitute into \( l^2 - 2m^2 + n^2 = 0 \): \( l^2 - 2m^2 + (2l+m)^2 = 0 \) \( l^2 - 2m^2 + 4l^2 + m^2 + 4lm = 0 \) \( 5l^2 + 4lm - m^2 = 0 \) Factorize: \( 5l^2 + 5lm - lm - m^2 = 0 \implies 5l(l+m) - m(l+m) = 0 \) \( (5l - m)(l + m) = 0 \). Cases: 1. \( m = 5l \): \( n = 2l + 5l = 7l \). Ratios \( (l, m, n) \sim (1, 5, 7) \). 2. \( m = -l \): \( n = 2l - l = l \). Ratios \( (l, m, n) \sim (1, -1, 1) \). Calculate \( \cos \theta \): Vectors are \( \vec{d_1} = (1, 5, 7) \) and \( \vec{d_2} = (1, -1, 1) \). Dot product: \( 1(1) + 5(-1) + 7(1) = 3 \). Magnitudes: \( \sqrt{1+25+49} = \sqrt{75} = 5\sqrt{3} \) and \( \sqrt{1+1+1} = \sqrt{3} \). \( \cos \theta = \frac{3}{5\sqrt{3} \cdot \sqrt{3}} = \frac{3}{15} = \frac{1}{5} \).
Step 4: Final Answer:

The value is \( \frac{1}{5} \).