Pair Of Straight Lines
The homogeneous second-degree equation 6x^2 + 11xy + 3y^2 = 0 represents a pair of straight lines through the origin; what is the acute angle between them?
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About This Question
- Subject
- mathematics
- Chapter
- coordinate geometry
- Topic
- pair of straight lines
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
tan−1(7/9)
A homogeneous equation ax^2 + 2hxy + by^2 = 0 always factors into two lines through the origin, and the angle between them follows from tan(theta) = 2 sqrt(h^2 - ab) / |a + b|. Here a = 6, b = 3, and 2h = 11 so h = 11/2. The sum of coefficients is a + b = 9, which fixes the denominator of the tangent expression. The numerator is 2 sqrt(h^2 - ab) = 2 sqrt((11/2)^2 - 6*3) = 2 sqrt(121/4 - 18) = 2 sqrt(49/4) = 2 * (7/2) = 7. Dividing by a + b = 9 gives tan(theta) = 7/9, so the acute angle between the pair of lines is tan^{-1}(7/9). Option tan^{-1}(5/9) misreads the discriminant h^2 - ab. Option tan^{-1}(5/3) ignores the denominator a + b. Option tan^{-1}(1) wrongly assumes perpendicular lines, which would require a + b = 0. The result tan^{-1}(7/9) is the acute angle since the ratio 7/9 is positive and less than 1, matching the discriminant-over-trace structure of the angle formula.
This hard difficulty mathematics question is from the chapter coordinate geometry, covering the topic of pair of straight lines. It appeared in the 2025 exam.
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