--- title: "Planes through line parallel to axes" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:planes-through-line-parallel-axes" source: https://www.bctengineer.com/q/semester_2/mathematics/planes-through-line-parallel-axes site: BCT Engineering --- # Planes through line parallel to axes Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2073 Bhadra, 5 marks; 2070 Magh, 5 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/planes-through-line-parallel-axes - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Find the equation of the plane through the line 2x+3y−5z = 4 and 3x−4y+5z = 6 and parallel to the coordinate axes. ## Short answer Plane through line: (2+3λ)x+(3−4λ)y+(−5+5λ)z=4+6λ. Setting coefficient of x to zero (λ=−2/3) gives 17y−25z=0. For y zero (λ=3/4): 17x−5z−34=0. For z zero (λ=1): 5x−y−10=0. ## Answer **Given line.** The line of intersection of 2x + 3y − 5z = 4 and 3x − 4y + 5z = 6. Any plane through it is (2x + 3y − 5z − 4) + λ(3x − 4y + 5z − 6) = 0, that is, (2 + 3λ)x + (3 − 4λ)y + (−5 + 5λ)z − (4 + 6λ) = 0. A plane is parallel to a coordinate axis when the coefficient of that variable is zero. #### Parallel to the x-axis 2 + 3λ = 0, λ = −2/3. Then the y-coefficient is 3 + 8/3 = 17/3, the z-coefficient is −5 − 10/3 = −25/3 and the constant is −(4 − 4) = 0. **17y − 25z = 0** #### Parallel to the y-axis 3 − 4λ = 0, λ = 3/4. The x-coefficient is 2 + 9/4 = 17/4, the z-coefficient is −5 + 15/4 = −5/4 and the constant is −(4 + 9/2) = −17/2. (17/4)x − (5/4)z − 17/2 = 0, so **17x − 5z − 34 = 0** #### Parallel to the z-axis −5 + 5λ = 0, λ = 1. The x-coefficient is 5, the y-coefficient is −1 and the constant is −10. **5x − y − 10 = 0** Each plane passes through the given line; for example, adding the two given planes (λ = 1) gives 5x − y = 10, which does not contain z. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is any plane through the line of intersection of the two planes written? A. (2x + 3y - 5z - 4)(3x - 4y + 5z - 6) = 0 B. (2x + 3y - 5z - 4) + lambda(3x - 4y + 5z - 6) = 0 C. (2x + 3y - 5z - 4) = lambda D. 2x + 3y - 5z - 4 + 3x - 4y + 5z - 6 = lambda Answer: B. A plane1 + lambda plane2 = 0 contains the intersection line for every lambda. (Key point: Family of planes) 2. When is such a plane parallel to a coordinate axis? A. The coefficient of that variable is zero B. The constant term is zero C. The coefficient of that variable is 1 D. The sum of coefficients is zero Answer: A. A plane Ax + By + Cz + D = 0 is parallel to the x-axis if A = 0, and similarly for the others. (Key point: Parallel to an axis) 3. What is the plane parallel to the x-axis? A. 17x - 5z - 34 = 0 B. 5x - y - 10 = 0 C. 17y - 25z = 0 D. 25y - 17z = 0 Answer: C. lambda = -2/3 makes the x-coefficient zero, giving 17y - 25z = 0. (Key point: Parallel to x-axis) 4. What is lambda and the plane parallel to the z-axis? A. lambda = 1, plane 5x - y - 10 = 0 B. lambda = -1, plane x - 7y + 2 = 0 C. lambda = 3/4, plane 17x - 5z - 34 = 0 D. lambda = 1, plane 5x + y - 10 = 0 Answer: A. Adding the two planes removes z, giving 5x - y = 10. 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