Lines (x−2)/1 = (y−3)/1 = (z−4)/−k and (x−1)/k = (y−4)/2 = (z−5)/1 are coplanar if k equals:

Lines (x−2)/1 = (y−3)/1 = (z−4)/−k and (x−1)/k = (y−4)/2 = (z−5)/1 are coplanar if k equals:

• A. 0 or −3
• B. 0 or 3
• C. 1 or −1
• D. 2 or −2

Answer: A) 0 or −3

Explanation: Lines are coplanar if the determinant of (x₂−x₁, y₂−y₁, z₂−z₁) and the two DRs is 0. Point diff: (1−2, 4−3, 5−4) = (−1, 1, 1). Det = (−1)(1 + 2k) − 1(1 − (−k²)) + 1(2 − k) = −1 − 2k − 1 − k² + 2 − k = −k² − 3k = 0. So k(k + 3) = 0. k = 0 or −3.