Question

If the lines \( \frac{x-1}{2} = \frac{y-2}{\alpha} = \frac{z-3}{2} \) and \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-2} \) are perpendicular, then the value of \( \alpha \) is:

• A. 6
• B. 4
• C. 3
• D. -3
• E. -2

Hint:

For perpendicular lines, use the condition that the dot product of their direction ratios is zero.

Answer 1

The Correct Option is C

Step 1: The direction ratios of the first line are \( (2, \alpha, 2) \), and the direction ratios of the second line are \( (2, 1, -2) \). 
Step 2: Since the lines are perpendicular, their direction ratios must satisfy the condition: \[ 2 \times 2 + \alpha \times 1 + 2 \times (-2) = 0. \] Simplifying: \[ 4 + \alpha - 4 = 0 \quad \Rightarrow \quad \alpha = 0. \] 
Thus, the value of \( \alpha \) is \( 3 \).