Question

Find an equation, or a set of equations, to describe the set of points that are equidistant from the points p(−9, 0, 0) and q(3, 0, 0). (enter your answers as a comma-separated list of equations.)

Answer 1

X = -3.  
The distance from p(-9, 0, 0) is
 d = sqrt((x+9)^2 + y^2 + z^2) 
 The distance from q(3,0,0) is
 d = sqrt((x-3)^2 + y^2 + z^2) 
 Let's set them equal to each other.
 sqrt((x+9)^2 + y^2 + z^2) = sqrt((x-3)^2 + y^2 + z^2)
  Square both sides, then simplify
 (x+9)^2 + y^2 + z^2 = (x-3)^2 + y^2 + z^2
 x^2 + 18x + 81 + y^2 + z^2 = x^2 - 6x + 9 + y^2 + z^2
 18x + 81 = - 6x + 9
 24x + 81 = 9
 24x = -72
 x = -3 
 So the desired equation is x = -3 which defines a plane.