Determine the point in yz-plane which is equidistant from three points

1.	Determine the point in yz-plane which is equidistant from three points A (2, 0 3) B (0, 3, 2) and C (0, 0, 1).
Answer» Since x-coordinate of every point in yz-plane is zero. Let P (0, y, z) be a point on the yz-plane such that PA = PB = PC. Now PA = PB ⇒ (0 – 2)² + (y – 0)² + (z – 3)²= (0 – 0)² + (y – 3)² + (z – 2)² , i.e. z – 3y = 0 and PB = PC ⇒ y² + 9 – 6y + z² + 4 – 4z = y² + z² + 1 – 2z , i.e. 3y + z = 6 Simplifying the two equating, we get y = 1, z = 3 Here, the coordinate of the point P are (0, 1, 3).

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

Answer»

Since x-coordinate of every point in yz-plane is zero. Let P (0, y, z) be a point on the yz-plane such that PA = PB = PC. Now

PA = PB ⇒ (0 – 2)² + (y – 0)² + (z – 3)²= (0 – 0)² + (y – 3)² + (z – 2)² , i.e. z – 3y = 0 and PB = PC

⇒ y² + 9 – 6y + z² + 4 – 4z = y² + z² + 1 – 2z , i.e. 3y + z = 6

Simplifying the two equating, we get y = 1, z = 3

Here, the coordinate of the point P are (0, 1, 3).