UPSC NDA I 2026 Mathematics Paper

Concept:A sphere concentric with $S$ has the same centre; if it passes through the origin, its radius is the distance from the centre to the origin.Explanation:The given sphere is$$x^2+y^2+z^2-4x-6y-12z+k=0.$$Its centre is obtained by completing the square:$$(x-2)^2+(y-3)^2+(z-6)^2 = 49-k.$$So the centre is $(2,3,6)$.The required sphere has the same centre $(2,3,6)$ and passes through the origin $(0,0,0)$.Therefore its radius is the distance between these points:$$r=\sqrt{(2-0)^2+(3-0)^2+(6-0)^2}=\sqrt{4+9+36}=\sqrt{49}=7.$$Answer:$$7$$