1. Identify the center of Sphere S:
The general equation of a sphere is given by x2+y2+z2+2ux+2vy+2wz+d=0, where the center is (−u,−v,−w).
Given equation: x2+y2+z2−4x−6y−12z+k=0
Comparing the coefficients:
2u=−4⟹u=−2
2v=−6⟹v=−3
2w=−12⟹w=−6
The center (h,k0,l) of sphere S is (2,3,6).
2. Determine the radius of the new sphere:
A sphere concentric with S will have the same center, (2,3,6).
Since the new sphere passes through the origin (0,0,0), its radius R is the distance between the center (2,3,6) and the origin (0,0,0).
Using the distance formula:
R=(2−0)2+(3−0)2+(6−0)2
R=22+32+62
R=4+9+36
R=49
R=7
Conclusion:
The radius of the required sphere is 7.
The correct option is (c) 7.
