--- title: "Sphere touching given sphere at a point" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:sphere-touching-given-sphere-point" source: https://www.bctengineer.com/q/semester_2/mathematics/sphere-touching-given-sphere-point site: BCT Engineering --- # Sphere touching given sphere at a point Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2068 Bhadra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/sphere-touching-given-sphere-point - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Find the equation which touches the sphere x²+y²+z²+2x−6y+1 = 0 at (1,2,−2) and passes through the point (1,−1,0). ## Short answer The required sphere has the form S0 + kT = 0 with tangent plane T: 2x - y - 2z - 4 = 0. Passing through (1,-1,0) yields k = 11, giving the equation x²+y²+z²+24x-17y-22z-43 = 0. ## Answer [Read as: find the equation of the sphere which touches the given sphere at (1, 2, −2) and passes through (1, −1, 0).] Any sphere touching S₀ at a point has the tangent plane T of S₀ at that point as the common tangent plane, so the required sphere is S₀ + kT = 0. S₀ = x² + y² + z² + 2x − 6y + 1 = 0. #### Check the point is on S₀ 1 + 4 + 4 + 2 − 12 + 1 = 0, so (1, 2, −2) lies on S₀. #### Tangent plane at (1, 2, −2) Use xx₁ + yy₁ + zz₁ + (x + x₁) − 3(y + y₁) + 1 = 0: ```text x + 2y − 2z + (x + 1) − 3(y + 2) + 1 = 0 2x − y − 2z − 4 = 0 ``` #### Sphere family ```text x² + y² + z² + 2x − 6y + 1 + k(2x − y − 2z − 4) = 0 ``` #### Pass through (1, −1, 0) ```text 1 + 1 + 0 + 2 + 6 + 1 + k(2 + 1 − 0 − 4) = 0 11 − k = 0 → k = 11 ``` #### Required sphere ```text x² + y² + z² + 2x − 6y + 1 + 11(2x − y − 2z − 4) = 0 ``` **Answer: x² + y² + z² + 24x − 17y − 22z − 43 = 0.** Check: at (1, −1, 0): 1 + 1 + 24 + 17 − 43 = 0. At (1, 2, −2): 1 + 4 + 4 + 24 − 34 + 44 − 43 = 0. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Any sphere touching S0 at a point shares which plane with S0? A. The plane through the centre B. The tangent plane at that point, so the sphere is S0 + kT = 0 C. The plane z = 0 D. The polar plane of the origin Answer: B. Spheres touching at a point have the same tangent plane there. (Key point: Family) 2. Tangent plane of S0 at (1, 2, -2) is: A. 2x - y - 2z - 4 = 0 B. 2x + y - 2z - 4 = 0 C. x - y - 2z - 4 = 0 D. 2x - y + 2z + 4 = 0 Answer: A. From xx1 + yy1 + zz1 + (x+x1) - 3(y+y1) + 1 = 0. (Key point: Tangent plane) 3. Passing through (1, -1, 0) gives: A. k = -11 B. k = 1 C. k = 11 D. k = 0 Answer: C. 11 - k = 0. (Key point: Determining k) 4. The required sphere is: A. x^2+y^2+z^2 + 24x - 17y + 22z - 43 = 0 B. x^2+y^2+z^2 + 24x - 17y - 22z - 43 = 0 C. x^2+y^2+z^2 + 2x - 6y + 1 = 0 D. x^2+y^2+z^2 + 13x - 7y - 22z - 43 = 0 Answer: B. 2 + 22 = 24, -6 - 11 = -17, -22, 1 - 44 = -43. (Key point: Sphere) ## Related in Mathematics - [Plane touches sphere, find point of contact](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aplane-touches-sphere-find-point) — 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 - [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 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 four points](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-four-points) — Mathematics, chapter 3 - [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 ## Related in other subjects - [Checking whether root locus passes through point P](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-point-p-dominant) — Control Systems, chapter 5 - [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 - [Gauss 3-point: 1/(x²+2x+1)](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Agauss-3pt-1-over-x2-2x-1) — Numerical Method, chapter 5, asked 3×