Consider the following equations of straight lines: Line L1 : 2x - 3y = 5 Line L2 : 3x + 2y = 8 Line L3 : 4x - 6y = 5 Line L4 : 6x - 9y = 6 Which one among the following is the correct statement? A. L1 is parallel to L2 and L1 is perpendicular to L2. B. L2 is parallel to L4 and L2 is perpendicular to L1. C. L2 is perpendicular to L4 and L1 is parallel to L3. D. L4 is perpendicular to L2 and L4 is parallel to L3.

GATE 2022 · General Aptitude · Lines · medium

Answer: Option D: L4 is perpendicular to L2 and L4 is parallel to L3.

1. Compute slopes of all four lines: L1: 2x - 3y = 5 => m1 = -2/(-3) = 2/3 L2: 3x + 2y = 8 => m2 = -3/2 L3: 4x - 6y = 5 => m3 = -4/(-6) = 2/3 L4: 6x - 9y = 6 => m4 = -6/(-9) = 2/3
2. Check parallel relationships: m1 = m3 = m4 = 2/3, so L1 || L3, L1 || L4, and L3 || L4. m2 = -3/2 differs from all others, so L2 is parallel to none of them.
3. Check perpendicular relationships: m1 × m2 = (2/3) × (-3/2) = -1 => L1 perp L2 (TRUE) m3 × m2 = (2/3) × (-3/2) = -1 => L3 perp L2 (TRUE) m4 × m2 = (2/3) × (-3/2) = -1 => L4 perp L2 (TRUE) No pair among L1, L3, L4 is perpendicular (same slope, product = (2/3)^2 != -1).
4. Evaluate each option: A: 'L1 || L2 AND L1 perp L2' — contradictory and false (m1 != m2, but that's good; the 'parallel' part is false anyway). B: 'L2 || L4 AND L2 perp L1' — L2 || L4 is FALSE (m2 = -3/2 != 2/3 = m4); reject. C: 'L2 perp L4 AND L1 || L3' — L2 perp L4 is TRUE, L1 || L3 is TRUE; both sub-claims hold. D: 'L4 perp L2 AND L4 || L3' — L4 perp L2 is TRUE, L4 || L3 is TRUE; both sub-claims hold.
5. Confirm option D: m4 × m2 = (2/3) × (-3/2) = -6/6 = -1 => L4 perp L2 TRUE m4 = m3 = 2/3 => L4 || L3 TRUE Both claims in D are satisfied.