Question

Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Answer 1

Let P (x, y, z) be the point that is equidistant from points A(1, 2, 3) and B(3, 2, -1).
Accordingly, PA = PB
⇒ PA2 = PB2
⇒(x-1)2+(y-2)2 +(z−3)2 = (x−3)2+(y−2)2 +(z+1)2
⇒ x2 - 2x + 1+ y2 - 4y+4+z2 - 6z + 9 = x2 - 6x+9+ y2 - 4y+ 4 + z2+2z+1
⇒ - 2x - 4y - 6z +14= - 6x - 4y+ 2z+14
⇒ - 2x - 6z + 6x - 2z = 0
⇒ 4x - 8z = 0
⇒x - 2z= 0
Thus, the required equation is x - 2z = 0.