Explanation
1. Recall the General Equation of a Sphere:
The general form is x2+y2+z2+2ux+2vy+2wz+d=0.
The center is given by (−u,−v,−w) and the radius r is calculated using:
r=u2+v2+w2−d
2. Map the Given Equation:
Equation: x2+y2+z2−4x−6y−12z+k=0
Comparing coefficients:
2u=−4⟹u=−2
2v=−6⟹v=−3
2w=−12⟹w=−6
d=k
3. Substitute into the Radius Formula:
We are given that the radius r=8.
8=(−2)2+(−3)2+(−6)2−k
Square both sides to eliminate the square root:
82=(−2)2+(−3)2+(−6)2−k
64=4+9+36−k
64=49−k
4. Solve for k:
k=49−64
k=−15
Conclusion:
The value of k is −15.
The correct option is (a) -15.
