Q.8 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 L3
(B) L2 is parallel to L4 and L2 is perpendicular to L1
(C) L3 is perpendicular to L4 and L3 is parallel to L2
(D) L4 is perpendicular to L2 and L4 is parallel to L3

L1 (2x – 3y = 5) and L3 (4x – 6y = 5) share the same slope, confirming they are parallel lines. No pairs among these lines are perpendicular. Option (D) is correct as L4 matches L3’s slope (parallel) but confirms perpendicularity with L2.

Slopes Calculation

Convert each equation to slope-intercept form y = mx + c, where m is the slope: m = -a/b for ax + by = c.

• L1: m₁ = 2/3
• L2: m₂ = -3/2
• L3: m₃ = 4/6 = 2/3
• L4: m₄ = 6/9 = 2/3

L1, L3, and L4 all have slope 2/3, so they are parallel to each other. L2’s slope -3/2 differs, and 2/3 × -3/2 = -1, making L2 perpendicular to L1, L3, and L4.

Option Analysis

Note: L4 is perpendicular to L2 (product = -1), making both claims in (D) valid.

Introduction to Parallel and Perpendicular Lines

Straight lines parallel perpendicular analysis is crucial for CSIR NET Life Sciences quantitative aptitude. Identify relations using slopes: equal slopes mean parallel lines; product of slopes = -1 means perpendicular. This question tests these rules on equations L1: 2x−3y=5, L2: 3x+2y=8, L3: 4x−6y=5, L4: 6x−9y=6.

Step-by-Step Slope Method

Rewrite in y = mx + c:

• L1, L3, L4: m = 2/3 (parallel family)
• L2: m = -3/2 (perpendicular to above)

Verification rules:

• Parallel: m₁ = m₂
• Perpendicular: m₁m₂ = -1

Why Option D Wins CSIR NET

L4 ⊥ L2 (2/3 × -3/2 = -1) and L4 ∥ L3 (same slope). Other options mix relations incorrectly. Practice similar for CSIR NET straight lines questions.

Keywords: straight lines parallel perpendicular, CSIR NET lines equations, L1 2x-3y=5 parallel L3, slope perpendicular product -1