<p>Given lines: \(x = ay + b,\; z = cy + d\) and \(x = a'y + b',\; z = c'y + d'\). These lines will be perpendicular to each other if:</p>
Step-by-Step Solution
Key Concept: Two lines in 3D given in symmetric form with parameter y have direction vectors (a, 1, c) and (a', 1, c'). They are perpendicular when their dot product equals zero: aa' + 1 + cc' = 0.
Step 1: Rewrite the lines in parametric form using y as parameter. Line 1: x = ay + b, y = y, z = cy + d has direction vector d_1 = (a, 1, c) Line 2: x = a'y + b', y = y, z = c'y + d' has direction vector d_2 = (a', 1, c') Step 2: Apply perpendicularity condition: d_1 · d_2 = 0 aa' + (1)(1) + cc' = 0 Step 3: The perpendicularity condition is: aa' + cc' + 1 = 0 ∴ Answer: B
Correct Answer: B