Question

Find an equation of the sphere with points p such that the distance from p to a(−2, 6, 3) is twice the distance from p to b(5, 2, −1).

Answer 1

Distance between p and a, PA is twice as distance between p and b, PB. a(â’2, 6, 3), b(5, 2, â’1), PA = 2PB 
Square root of [(x + 2)^2 + (y - 6)^2 + (z - 3)^2] = 2 x Square root of [(x - 5)^2
+ (y - 2)^2 + (z + 1)^2] 
 Removing the square roots,
  [(x + 2)^2 + (y - 6)^2 + (z - 3)^2] = 4[(x - 5)^2 + (y - 2)^2 + (z + 1)^2]
 x^2 + y^2 + z^2 + 4x - 12y - 6z + 4 + 36 + 9 = 4(x^2 + y^2 + z^2 - 10x - 4y +
2z + 25 + 4 +1) 
 x^2 + y^2 + z^2 + 4x - 12y - 6z + 49 = 4x^2 + 4y^2 + 4z^2 - 40x -16y + 8z +
120 
Hence the equation of sphere = 3x^2 + 3y^2 + 3z^2 - 44x - 4y + 14z + 71