Straight Lines
Statement: the lines 2x - 3y + 5 = 0 and 3x + 2y - 7 = 0 are mutually perpendicular; which option correctly evaluates this claim and its reasoning?
Select the correct option:
Solution
True, because the product of their slopes equals -1
Two lines are perpendicular precisely when the product of their slopes is -1, a criterion that converts the geometric notion of a right angle into a clean algebraic test. The first line 2x - 3y + 5 = 0 rearranges to y = (2/3)x + 5/3, giving slope m_1 = 2/3. The second line 3x + 2y - 7 = 0 rearranges to y = -(3/2)x + 7/2, giving slope m_2 = -3/2. Their product is (2/3)(-3/2) = -1, so the lines are indeed perpendicular and the reasoning via slope product is correct. The option citing equal slopes is wrong because the slopes 2/3 and -3/2 are clearly unequal. The option calling the slopes reciprocals ignores the required sign change for perpendicularity. The option claiming the lines are parallel is false since parallel lines need equal slopes. This applies the standard JEE Advanced perpendicularity condition. Plausibility check: the slopes are negative reciprocals, the exact signature of perpendicular lines, confirming the verdict.
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About This Question
- Subject
- mathematics
- Chapter
- coordinate geometry
- Topic
- straight lines
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
True, because the product of their slopes equals -1
Two lines are perpendicular precisely when the product of their slopes is -1, a criterion that converts the geometric notion of a right angle into a clean algebraic test. The first line 2x - 3y + 5 = 0 rearranges to y = (2/3)x + 5/3, giving slope m_1 = 2/3. The second line 3x + 2y - 7 = 0 rearranges to y = -(3/2)x + 7/2, giving slope m_2 = -3/2. Their product is (2/3)(-3/2) = -1, so the lines are indeed perpendicular and the reasoning via slope product is correct. The option citing equal slopes is wrong because the slopes 2/3 and -3/2 are clearly unequal. The option calling the slopes reciprocals ignores the required sign change for perpendicularity. The option claiming the lines are parallel is false since parallel lines need equal slopes. This applies the standard JEE Advanced perpendicularity condition. Plausibility check: the slopes are negative reciprocals, the exact signature of perpendicular lines, confirming the verdict.
This easy difficulty mathematics question is from the chapter coordinate geometry, covering the topic of straight lines. It appeared in the 2025 exam.
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