Question

If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then \( (\alpha, \beta) \) is:

• A. \( (-1,-1) \)
• B. \( (1,-1) \)
• C. \( (1,3) \)
• D. \( (1,1) \)

Hint:

For perpendicularity in 3D, use the dot product condition, and for parallelism, equate the ratios of direction cosines.

Answer 1

The Correct Option is B

Step 1: Using perpendicularity condition Two lines are perpendicular if: \[ \alpha_1 \alpha_2 + \beta_1 \beta_2 + \gamma_1 \gamma_2 = 0. \] Solving with the given ratios: \[ 1(-1) + a(2) + \beta(1) = 0. \] Step 2: Using parallel condition Since the lines are parallel, \[ \alpha = 1, \quad \beta = -1. \]