--- title: "Centre and radius of circle of sphere section" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:centre-radius-circle-sphere-section" source: https://www.bctengineer.com/q/semester_2/mathematics/centre-radius-circle-sphere-section site: BCT Engineering --- # Centre and radius of circle of sphere section Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2072 Magh, 5 marks; 2070 Magh, 5 marks; 2067 Mangsir, 5 marks (asked 3 times) - Video: [Class 45 || Find center and radius of circle || Given sphere and plane || BA | BSc 3D Math](https://www.youtube.com/watch?v=h1e2xtmqVI0) — AC Prashant - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/centre-radius-circle-sphere-section - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Find the centre and radius of the circle in which the sphere x²+y²+z²−8x+4y+8z−45 = 0 is cut by the plane x−2y+2z = 3. ## Short answer Sphere centre (4,−2,−4), radius 9. Distance to plane x−2y+2z=3 is 1. Circle radius is √(81−1)=4√5. Centre is foot of perpendicular from (4,−2,−4): (13/3, −8/3, −10/3). ## Answer **Sphere:** x² + y² + z² − 8x + 4y + 8z − 45 = 0. Comparing with x² + y² + z² + 2ux + 2vy + 2wz + d = 0: u = −4, v = 2, w = 4, d = −45. ```text Centre C = (−u, −v, −w) = (4, −2, −4) Radius R = √(16 + 4 + 16 + 45) = √81 = 9 ``` **Plane:** x − 2y + 2z = 3, normal (1, −2, 2), |n| = 3. #### Distance of C from the plane ```text p = |4 − 2(−2) + 2(−4) − 3| / 3 = |4 + 4 − 8 − 3|/3 = 1 ``` #### Radius of the circle ```text r = √(R² − p²) = √(81 − 1) = √80 = 4√5 ≈ 8.94 ``` #### Centre of the circle It is the foot of the perpendicular from C on the plane. The perpendicular is (x − 4)/1 = (y + 2)/(−2) = (z + 4)/2 = t, so (4 + t, −2 − 2t, −4 + 2t). Put in the plane: ```text (4 + t) − 2(−2 − 2t) + 2(−4 + 2t) = 3 4 + t + 4 + 4t − 8 + 4t = 3 9t = 3 → t = 1/3 ``` Centre = (4 + 1/3, −2 − 2/3, −4 + 2/3) = (13/3, −8/3, −10/3). Check: 13/3 + 16/3 − 20/3 = 3 ✓. **Answer: centre (13/3, −8/3, −10/3), radius 4√5.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the centre and radius of x^2+y^2+z^2-8x+4y+8z-45 = 0? A. Centre (4, -2, -4), radius 9 B. Centre (-4, 2, 4), radius 9 C. Centre (4, -2, -4), radius 3 D. Centre (8, -4, -8), radius 9 Answer: A. u = -4, v = 2, w = 4, so centre (4, -2, -4) and R = sqrt(16 + 4 + 16 + 45) = 9. (Key point: Sphere centre and radius) 2. What is the distance from the centre to x - 2y + 2z = 3? A. 3 B. 0 C. 1 D. 9 Answer: C. |4 + 4 - 8 - 3|/3 = 1. (Key point: Distance to plane) 3. What is the radius of the circle? A. 4 sqrt(5) B. sqrt(82) C. 8 D. sqrt(79) Answer: A. r = sqrt(81 - 1) = sqrt(80) = 4 sqrt(5). (Key point: Circle radius) 4. What is the circle's centre? A. (4, -2, -4) B. (13/3, -8/3, -10/3) C. (1/3, -2/3, 2/3) D. (11/3, -4/3, -14/3) Answer: B. It is the foot of the perpendicular from C, at t = 1/3 on (4 + t, -2 - 2t, -4 + 2t). (Key point: Circle centre) ## Related in Mathematics - [Sphere through circle meeting plane in circle of radius 3](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-circle-meeting-plane) — Mathematics, chapter 3 - [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 - [Centre and radius of a circle on sphere](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acentre-radius-circle-sphere) — Mathematics, chapter 3, asked 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 - [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 - [Equation and length of axes of x²−4xy−2y²+10x+4y=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-x2-4xy-2y2-equation-axes) — Mathematics I, chapter 3 - [Midpoint circle: r=8, centre (10,10)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r8-center-10-10) — Computer Graphics, chapter 2, asked 2× - [Midpoint circle: r=6, centre (20,10)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r6-center-20-10) — Computer Graphics, chapter 2, asked 2× - [Midpoint circle: r=7, centre (10,15)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r7-center-10-15) — Computer Graphics, chapter 2