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

Given: Points (2, 1), (5, - 8) and (2, - 9).

To find: The centre of the circle.

Formula Used:

The distance between the points (x₁, y₁) and (x₂, y₂) is:

Distance

Explanation:

Coordinates of points on a circle are A(2,1), B(5, - 8) and C(2, - 9).
Let the coordinates of the centre of the circle be O(x, y)
Using distance formula

Since the distance of the points A, B and C will be equal from the centre, therefore
⇒ OA = OB

On squaring both sides, we get
⇒ x² + 4 - 4x + y² + 1 - 2y = x² + 25 - 10x + y² + 64 + 16y
⇒ 6x - 18y - 84 = 0
⇒ x - 3y - 14 = 0 - - - - - - - - - - - - - (1)
Similarly, OC = OB
⇒ x² + 4 - 4x + y² + 81 + 18y = x² + 25 - 10x + y² + 64 + 16y
⇒ 6x - 2y - 4 = 0
⇒ 3x - y - 2 = 0 - - - - - - - - - - - - - (2)
By solving equations (1) and (2), we get x = - 1, y = - 5
So, the coordinates of the centre of the circle is ( - 1, - 5).
Radius of the circle = OA =
=
= √45
= 3√5 units