Answer:
(x -26/3)² +(y -2)² +(z +4)² = 436/9
Explanation:
For P = (x, y, z), the distance to point A is ...
dA = √((x+2)² +(y-6)² +(z-4)²)
and the distance to point B is ...
dB = √((x-6)² +(y-3)² +(z+2)²)
We want the former to be twice the latter, so ...
dA = 2dB
√((x+2)² +(y-6)² +(z-4)²) = 2√((x-6)² +(y-3)² +(z+2)²)
Squaring both sides gives ...
(x+2)² +(y-6)² +(z-4)² = 4((x-6)² +(y-3)² +(z+2)²)
Subtracting the left side, we are left with ...
4(x-6)²-(x+2)² +4(y-3)² -(y-6)² +4(z+2)² -(z-4)² = 0
Expanding gives us the general form equation of the sphere:
3x² -52x +3y² -12y +3z² +24z +140 = 0
We can divide by 3 and complete the squares to get the standard-form equation:
(x -26/3)² +(y -2)² +(z +4)² = 436/9