Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).

Let the centre of the circle be O(x, y). The points on the circle are A(5, -8), B(2, -9) and C(2, 1).
OA
OB
OC ⇒ OA = OB = OC = radius

⇒ (5 - x) ² + (8 + y) ² = (2 - x) ² + (9 + y) ²
⇒ 25 – 10x + x ² + 64 + 16y + y ² = 4 - 4x + x ² + 81 + 18y + y ²
⇒ 89 – 10x + 16y = 85 - 4x + 18y
⇒ – 6x - 2y = - 4
⇒ 3x + y = 2 ………………………..(i)
⇒ (2 - x) ² + (9 + y) ² = (2 - x) ² + (1 - y) ²
⇒ 4 – 4x + x ² + 81 + 18y + y ² = 4 - 4x + x ² + 1 - 2y + y ²
⇒ 81 + 18y = 1 - 2y
⇒ 20y = -80
⇒ y = -4 ………………………..(ii)
Substituting y = -4 in (i)
⇒ 3x - 4 = 2
⇒ 3x = 6
⇒ x = 2
Therefore the centre of the circle is (2, -4).