Given that we need to find the centre of the circle passing through O(0,0), A(4,0) and B(0, - 6).

Let us find the lengths of the triangle OAB.

We know that the distance between the points (x₁,y₁) and (x₂,y₂) is .

⇒

⇒

⇒ OA = 4

⇒

⇒

⇒ OB = 6

⇒

⇒

⇒ AB = √52

Consider OA² + OB²,

⇒ OA² + OB² = 4² + 6²

⇒ OA² + OB² = 16 + 36

⇒ OA² + OB² = 52

⇒

⇒ OA² + OB² = AB²

We got triangle OAB is a right-angled triangle with AB as the hypotenuse.

We know that the circumcentre of the right-angled triangle is the midpoint of the hypotenuse.

⇒ Centre =

⇒ Centre = (2, - 3)

∴The coordinates of the centre is (2, - 3).