<p>The two lines <em>x = ay + b, z = cy + d</em> and <em>x = a'y + b', z = c'y + d'</em> are perpendicular to each other if</p>
<p>\(aa' + cc' = -1\)</p>
<p>\(aa' + cc' = 1\)</p>
<p>\(\dfrac{a}{a'} + \dfrac{c}{c'} = -1\)</p>
<p>\(\dfrac{a}{a'} + \dfrac{c}{c'} = 1\)</p>
Step-by-Step Solution
Key Concept: Two lines in 3D given in parametric form with y as parameter are perpendicular when their direction vectors have dot product equal to zero. Convert each line to standard form and extract direction vectors.
Step 1: Convert lines to parametric form with parameter t = y. Line 1: x = at + b, y = t, z = ct + d → Direction vector d_1 = (a, 1, c) Line 2: x = a't + b', y = t, z = c't + d' → Direction vector d_2 = (a', 1, c') Step 2: Apply perpendicularity condition: d_1 · d_2 = 0 (a, 1, c) · (a', 1, c') = 0 aa' + 1(1) + cc' = 0 Step 3: Simplify to get the perpendicularity condition. ∴ Answer: aa' + cc' + 1 = 0
Correct Answer: A