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 + (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.