Coplanarity Of Two Lines
Examine whether the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-3)/4 = (z-4)/5 are coplanar.
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About This Question
- Subject
- mathematics
- Chapter
- three dimensional geometry
- Topic
- coplanarity of two lines
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
They are coplanar
The deciding criterion is the coplanarity test using a scalar triple product: two lines through points a_1 and a_2 with directions b_1 and b_2 are coplanar if and only if (a_2 - a_1) \cdot (b_1 \times b_2) = 0. This determinant condition is the standard JEE Advanced test. Here a_1 = (1, 2, 3), a_2 = (2, 3, 4) so a_2 - a_1 = (1, 1, 1); b_1 = (2, 3, 4) and b_2 = (3, 4, 5). The cross product b_1 \times b_2 = (3\cdot5 - 4\cdot4,\ 4\cdot3 - 2\cdot5,\ 2\cdot4 - 3\cdot3) = (-1, 2, -1). The triple product is (1)(-1) + (1)(2) + (1)(-1) = -1 + 2 - 1 = 0, so the lines are coplanar. They are not skew, since skewness requires a nonzero triple product. They are not parallel, because b_1 and b_2 are not proportional. They are not identical, since the points differ and directions differ. This uses the determinant coplanarity theorem. Plausibility check: a zero scalar triple product geometrically means the connecting vector lies in the plane of the two directions, exactly the coplanarity condition.
This hard difficulty mathematics question is from the chapter three dimensional geometry, covering the topic of coplanarity of two lines. It appeared in the 2025 exam.
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