Find the equation of the set of points which are equidistant from the

1.	Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3,2,-1).
Answer» Let A (1, 2, 3) and B (3, 2, -1) be the given points. Let P(x, y, z) be any point equidistant from A and B, then PA = PB (i.e., 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 + 2z + 1 ⇒ x²- 6x + 9 + y²- 4y + 4 + z² + 2z + 1 ⇒ 4x – 8z = 0 ⇒ x – 2z = 0 is the required equation.

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

Answer»

Let A (1, 2, 3) and B (3, 2, -1) be the given points.

Let P(x, y, z) be any point equidistant from A and B, then

PA = PB (i.e., 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 + 2z + 1

⇒ x²- 6x + 9 + y²- 4y + 4 + z² + 2z + 1

⇒ 4x – 8z = 0

⇒ x – 2z = 0 is the required equation.