Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

Given: Points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1)

To find: the point on yz-plane which is equidistant from the points

As we know x = 0 in yz-plane.

Let Q(0, y, z) any point in yz-plane

According to the question:

QA = QB = QC

⇒ QA² = QB² = QC²

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

The distance between Q(0, y, z) and A(2, 0, 3) is QA,

The distance between Q(0, y, Z) and B(0, 3, 2) is QB,

Distance between Q(0, y, z) and C(0, 0, 1) is QC,

As QA² = QB²

4 + (z – 3)²+ y² = (z – 2)² + (y – 3)²

⇒ z²+ 9 – 6z + y² + 4 = z²+ 4 – 4z + y² + 9 – 6y

⇒ – 6z = – 4z – 6y

⇒ 6y – 6z + 4z = 0

⇒ 6y – 2z = 0

⇒ 6y = 2z

⇒ z = 3y…………………(1)

As QA² = QC²

4 + (z – 3)²+ y² = (z – 1)² + y²

⇒ z²+ 9 – 6z + y² + 4 = z²+ 1 – 2z + y²

⇒ 13 – 6z = 1 – 2z

⇒ 13 – 1 = 6z – 2z

⇒ 4z = 12

⇒ z = 3

Put the value of z from (1):

⇒ y = 1

Hence point Q(0, 1, 3) in yz-plane is equidistant from A, B and C