--- title: "Sphere through circle of two spheres, centre on a plane" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:sphere-through-circle-two-spheres" source: https://www.bctengineer.com/q/semester_2/mathematics/sphere-through-circle-two-spheres site: BCT Engineering --- # Sphere through circle of two spheres, centre on a plane Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2068 Magh, 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-through-circle-two-spheres - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Show that the equation of the sphere through the circle x²+y²+z²−2x−3y+4z+8 = 0 and x²+y²+z²+4x+5y−6z+2 = 0 and having its centre on the plane 4x−5y−z−3 = 0 is x²+y²+z²+7x+9y−11z−1 = 0. [First two equations are the two spheres whose intersection is the circle, as read from scan] ## Short answer Using the sphere pencil S1 + kS2 = 0 and imposing the centre lies on 4x-5y-z-3=0 gives k = -3. Simplifying yields x²+y²+z²+7x+9y-11z-1 = 0. ## Answer [The two given equations are the two spheres whose common circle is meant (as read from the scan). The given answer is verified below.] Let S₁ = x² + y² + z² − 2x − 3y + 4z + 8 and S₂ = x² + y² + z² + 4x + 5y − 6z + 2. #### Family of spheres through the circle S₁ + kS₂ = 0: ```text (1+k)(x²+y²+z²) + (4k−2)x + (5k−3)y + (4−6k)z + (8+2k) = 0 ``` Centre = ( (1−2k)/(1+k), (3−5k)/(2(1+k)), (3k−2)/(1+k) ). #### Centre on the plane 4x − 5y − z − 3 = 0 Multiply by 2(1 + k): ```text 8(1−2k) − 5(3−5k) − 2(3k−2) − 6(1+k) = 0 8 − 16k − 15 + 25k − 6k + 4 − 6 − 6k = 0 −9 − 3k = 0 → k = −3 ``` #### The sphere S₁ − 3S₂ = 0: ```text −2(x²+y²+z²) − 14x − 18y + 22z + 2 = 0 ``` Divide by −2: **x² + y² + z² + 7x + 9y − 11z − 1 = 0**, which is the required result. Check: the centre is (−7/2, −9/2, 11/2); 4(−7/2) − 5(−9/2) − 11/2 − 3 = −14 + 22.5 − 5.5 − 3 = 0. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Spheres through the intersection circle of S1 = 0 and S2 = 0 are: A. S1 S2 = k B. S1 + kS2 = 0 C. S1 - S2 = 0 only D. S1 + k = 0 Answer: B. Every member passes through S1 = S2 = 0. (Key point: Family) 2. How is k determined? A. By passing through the origin B. By making the radius 1 C. By requiring the centre to lie on 4x - 5y - z - 3 = 0 D. By setting 1 + k = 0 Answer: C. Gives -9 - 3k = 0. (Key point: Centre condition) 3. k equals: A. -3 B. 3 C. -9 D. 1/3 Answer: A. -9 - 3k = 0. (Key point: Value) 4. Dividing S1 - 3S2 by -2 gives: A. x^2+y^2+z^2 - 7x - 9y + 11z - 1 = 0 B. x^2+y^2+z^2 + 7x + 9y - 11z - 1 = 0 C. x^2+y^2+z^2 + 7x + 9y - 11z + 1 = 0 D. 2(x^2+y^2+z^2) + 7x + 9y - 11z - 1 = 0 Answer: B. -2(...) - 14x - 18y + 22z + 2 divided by -2. (Key point: Result) ## Related in Mathematics - [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 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 - [Plane through intersection, perpendicular to plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aplane-through-intersection-perpendicular-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× - [Sphere with given circle as great circle](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-given-circle-as-great) — Mathematics, chapter 3, asked 2× - [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× ## Related in other subjects - [Centre, axes of ellipse 2x²+3y²−4x−12y+13=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-2x2-3y2-centre-axes) — Mathematics I, chapter 3, asked 7× - [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 - [Simplex method max:Z=4x+3y](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-4x-3y) — Mathematics III, chapter 5, asked 17× - [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×