--- title: "Sphere through circle meeting plane in circle of radius 3" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:sphere-through-circle-meeting-plane" source: https://www.bctengineer.com/q/semester_2/mathematics/sphere-through-circle-meeting-plane site: BCT Engineering --- # Sphere through circle meeting plane in circle of radius 3 Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2070 Bhadra, 5 marks - Video: [Obtain equation of the sphere which passes through the circle x^2+y^2=4, z=0 and is cut by the plane](https://www.youtube.com/watch?v=QCBVz59lbcg) — Super Hundred Entrancepedia Institute - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/sphere-through-circle-meeting-plane - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Find the equation of the sphere through the circle x²+y² = 4, z = 0 and is intersected by the plane x+2y+2z = 0 in a circle of radius 3. ## Short answer Family x^2+y^2+z^2-4+kz=0 has centre (0,0,-k/2). Distance to plane x+2y+2z=0 is |k|/3. From section radius 3, k^2=36. Spheres: x^2+y^2+z^2±6z-4=0. ## Answer **Given.** Circle: x² + y² = 4, z = 0. The sphere also meets the plane x + 2y + 2z = 0 in a circle of radius 3. #### Family of spheres through the circle The circle lies on the sphere x² + y² + z² = 4 and on the plane z = 0, so every sphere through it is x² + y² + z² − 4 + kz = 0. Centre: (0, 0, −k/2). Radius²: R² = 4 + k²/4. #### Section by the plane x + 2y + 2z = 0 Distance of the centre from the plane: ```text p = |0 + 0 + 2(−k/2)| / √(1 + 4 + 4) = |k| / 3 ``` The circle of section has radius 3, so ```text R² − p² = 9 4 + k²/4 − k²/9 = 9 (5/36) k² = 5 k² = 36 → k = 6 or k = −6 ``` #### Spheres - k = 6: **x² + y² + z² + 6z − 4 = 0** - k = −6: **x² + y² + z² − 6z − 4 = 0** Check for k = 6: centre (0, 0, −3), R² = 13, p = 2, R² − p² = 9 ✓. **Answer: x² + y² + z² ± 6z − 4 = 0.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which family contains all spheres through the circle x^2+y^2 = 4, z = 0? A. x^2+y^2+z^2 - 4 + kz = 0 B. x^2+y^2+z^2 - 4 + k = 0 C. x^2+y^2 - 4 + kz = 0 D. x^2+y^2+z^2 + kz = 4k Answer: A. S + kP with S the sphere x^2+y^2+z^2 = 4 and P the plane z = 0. (Key point: Family of spheres) 2. For that family, what are the centre and R^2? A. (0, 0, k/2), 4 - k^2/4 B. (0, 0, -k/2), 4 + k^2/4 C. (0, 0, -k), 4 + k^2 D. (0, 0, -k/2), 4 - k^2/4 Answer: B. Completing the square in z gives centre (0, 0, -k/2) and R^2 = 4 + k^2/4. (Key point: Centre and radius) 3. If a plane at distance p from the centre cuts a circle of radius 3, which relation holds? A. R^2 + p^2 = 9 B. R - p = 3 C. R^2 - p^2 = 9 D. p^2 - R^2 = 9 Answer: C. Radius of the section circle squared = R^2 - p^2. (Key point: Section radius relation) 4. Which k values result from 4 + k^2/4 - k^2/9 = 9? A. k = 6 or k = -6 B. k = 6 only C. k = 36 D. k = 3 or k = -3 Answer: A. (5/36)k^2 = 5 gives k^2 = 36, so both signs are valid. (Key point: Values of k) ## Related in Mathematics - [Sphere through circle of two spheres, centre on a plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-circle-two-spheres) — Mathematics, chapter 3 - [Sphere through circle touching a plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-circle-touching-plane) — Mathematics, chapter 3 - [Centre and radius of circle of sphere section](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acentre-radius-circle-sphere-section) — Mathematics, chapter 3, asked 3× - [Tangent plane to sphere through line](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Atangent-plane-sphere-through-line) — Mathematics, chapter 3, asked 3× - [Sphere with circle as great circle](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-circle-as-great-circle) — Mathematics, chapter 3, asked 3× - [Sphere with given circle as great circle](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-given-circle-as-great-2) — Mathematics, chapter 3, asked 2× ## Related in other subjects - [Flux through sphere, two line charges](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aflux-leaving-sphere-two-line-charges) — Electromagnetism, chapter 2 - [Conduction through hollow sphere](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ahollow-sphere-heat-flow) — Heat Transfer, chapter 2 - [Heat flow through composite plane wall, three layers](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Acomposite-plane-wall-three-layers) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Plane through three points and distance](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aplane-surface-three-points) — Computer Aided Design, chapter 4