--- title: "Coplanar lines and common point" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:coplanar-lines-common-point" source: https://www.bctengineer.com/q/semester_2/mathematics/coplanar-lines-common-point site: BCT Engineering --- # Coplanar lines and common point Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2073 Magh, 5 marks - Video: [COPLANAR LINES || INTERSECTION POINT OF TWO LINE || STRAIGHT LINE|| CLASS 12](https://www.youtube.com/watch?v=gRSInLTxWSo) — Masoom Tutorials - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/coplanar-lines-common-point - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Show that the lines (x−1)/2 = (y+1)/(−3) = (z+10)/8 and (x−4)/1 = (y+3)/(−4) = (z+1)/7 are coplanar. Find their common point. ## Short answer AB=(3,-2,9), d1=(2,-3,8), d2=(1,-4,7). Determinant=0 proves coplanarity. Param equation t=2, s=1 satisfy all coordinates; intersection point is (5,-7,6). ## Answer **Lines.** - L₁: (x − 1)/2 = (y + 1)/(−3) = (z + 10)/8, through A(1, −1, −10), direction (2, −3, 8). - L₂: (x − 4)/1 = (y + 3)/(−4) = (z + 1)/7, through B(4, −3, −1), direction (1, −4, 7). #### Coplanarity test Two lines are coplanar if the determinant of AB, d₁, d₂ is zero. AB = (3, −2, 9). ```text | 3 −2 9 | | 2 −3 8 | = 3(−21 + 32) + 2(14 − 8) + 9(−8 + 3) | 1 −4 7 | = 3(11) + 2(6) + 9(−5) = 33 + 12 − 45 = 0 ``` The determinant is zero, so the lines are coplanar. #### Common point General points: L₁: (1 + 2t, −1 − 3t, −10 + 8t); L₂: (4 + s, −3 − 4s, −1 + 7s). Equate: ```text 1 + 2t = 4 + s → 2t − s = 3 −1 − 3t = −3 − 4s → −3t + 4s = −2 ``` From the first, s = 2t − 3. Put in the second: −3t + 8t − 12 = −2, so 5t = 10, t = 2 and s = 1. Check the third coordinate: −10 + 8(2) = 6 and −1 + 7(1) = 6 ✓. Point: x = 1 + 4 = 5, y = −1 − 6 = −7, z = 6. **Answer: the lines are coplanar and meet at (5, −7, 6).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What determinant is evaluated? A. The determinant of d1, d2, and the origin B. The determinant of AB, d1, d2 C. The determinant of A, B and d1 + d2 D. The determinant of d1, d2 only Answer: B. Lines are coplanar if AB, d1 and d2 are coplanar vectors. (Key point: Coplanarity test) 2. What is the value of the determinant? A. 0 B. -12 C. 45 D. 33 Answer: A. 33 + 12 - 45 = 0. (Key point: Determinant value) 3. What are t and s for the common point? A. t = 1, s = 2 B. t = 2, s = 2 C. t = 2, s = 1 D. t = 3, s = 1 Answer: C. From 2t - s = 3 and -3t + 4s = -2. (Key point: Parameters) 4. What is the common point? A. (5, 7, 6) B. (3, -4, 6) C. (5, -7, -6) D. (5, -7, 6) Answer: D. Putting t = 2 in L1 gives (5, -7, 6). (Key point: Common point) ## Close variants set in other exams - [Coplanar lines with point of intersection](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-point-intersection-2) — Mathematics, chapter 3, asked 3× ## Related in Mathematics - [Coplanar lines and point of intersection](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-point-intersection) — Mathematics, chapter 3, asked 2× - [Coplanar lines, intersection point and plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-intersection-point-plane) — Mathematics, chapter 3, asked 2× - [Coplanar lines and plane containing them](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-plane-containing-them-2) — Mathematics, chapter 3, asked 3× - [Coplanar lines and plane containing them](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-plane-containing-them) — Mathematics, chapter 3, asked 2× - [Prove lines coplanar](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aprove-lines-coplanar) — Mathematics, chapter 3 - [Lines intersect: point and plane through them](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Alines-intersect-point-plane-through) — Mathematics, chapter 3 ## Related in other subjects - [Gradient P to Q from common point R](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Agradient-pq-common-point-r) — Surveying II, chapter 2 - [IC, Q-point, load line, IB = 10 µA](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ace-ic-q-point-ib-10ua) — Basic Electronics Engineering, chapter 3 - [Annual bleaching powder 10 MLD; break point](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ableaching-s71) — Water Supply Engineering, chapter 6 - [Max BM at 10 m: UDL and two point loads](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-udl-4m-two-10kn-25m) — Theory of Structures I, chapter 5