Question

If \( l, m, n \) are the direction cosines of a line that is perpendicular to the lines having the direction ratios \( (1,2,-1) \) and \( (-2,1,1) \), then \( (l+m+n)^2 \) =

• A. \( \frac{1}{20} \)
• B. \( \frac{9}{5} \)
• C. \( \frac{1}{5} \)
• D. \( \frac{3}{20} \)

Hint:

To find the direction cosines of a line perpendicular to two given lines, solve the system of equations using perpendicularity conditions and apply the direction cosine equation \( l^2 + m^2 + n^2 = 1 \).

Answer 1

The Correct Option is B

Step 1: Condition for perpendicularity
If a line is perpendicular to two given lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \), then its direction cosines \( (l, m, n) \) satisfy: \[ a_1 l + b_1 m + c_1 n = 0, \] \[ a_2 l + b_2 m + c_2 n = 0. \] For the given lines: 1st line: \( (1,2,-1) \), so \[ 1l + 2m - 1n = 0. \] 2nd line: \( (1,-2,1) \), so \[ 1l - 2m + 1n = 0. \] Step 2: Solving for \( l, m, n \)
Adding the two equations: \[ (1+1)l + (2-2)m + (-1+1)n = 0. \] \[ 2l = 0 \Rightarrow l = 0. \] Substituting \( l = 0 \) in the first equation: \[ 2m - n = 0 \Rightarrow n = 2m. \] Using the direction cosine condition: \[ l^2 + m^2 + n^2 = 1. \] \[ 0^2 + m^2 + (2m)^2 = 1. \] \[ m^2 + 4m^2 = 1. \] \[ 5m^2 = 1 \Rightarrow m^2 = \frac{1}{5}. \] Step 3: Computing \( (l+m+n)^2 \)
\[ (l+m+n)^2 = (0 + m + 2m)^2 = (3m)^2 = 9m^2. \] \[ = 9 \times \frac{1}{5} = \frac{9}{5}. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{\frac{9}{5}}. \]