Straight Lines
Two lines given by 3x - 4y + 7 = 0 and 6x - 8y - 5 = 0 are parallel; what is the perpendicular distance between this pair of parallel lines?
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About This Question
- Subject
- mathematics
- Chapter
- coordinate geometry
- Topic
- straight lines
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
19/10
The distance between two parallel lines requires identical coefficients of x and y, after which the gap is the difference of constants divided by the norm of the normal vector. Rewriting 6x - 8y - 5 = 0 by dividing by 2 gives 3x - 4y - 5/2 = 0, so both lines share the normal (3, -4) with magnitude sqrt(9 + 16) = 5. The distance formula |c1 - c2| / sqrt(a^2 + b^2) then gives |7 - (-5/2)| / 5 = |19/2| / 5 = 19/10. Option 12/10 wrongly uses |7 - (-5)|/... without halving the second equation. Option 2/5 comes from forgetting to normalize the second equation entirely. Option 7/5 results from dropping the second constant and using only 7/5. This is the canonical JEE Advanced parallel-line distance computation, hinging on first matching coefficients. Plausibility check: 19/10 = 1.9 is a modest positive separation, consistent with two nearly coincident lines whose constants differ by 19/2 once normalized, so the magnitude is geometrically reasonable.
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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