Using distance formula show that (3, 3) is the centre of the circle passing through the points (6, 2), (0, 4) and (4, 6). Find the radius of the circle.

Given that circle passes through the points A(6, 2), B(0, 4), C(4, 6).

Let us assume O(x, y) be the centre of the circle.

We know that distance from the centre to any point on h circle is equal.

So, OA = OB = OC

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

Now,

⇒ OA = OB

⇒ OA² = OB²

⇒ (x - 6)² + (y - 2)² = (x - 0)² + (y - 4)²

⇒ x² - 12x + 36 + y² - 4y + 4 = x² + y² - 8y + 16

⇒ 12x - 4y = 24

⇒ 3x - y = 6 ..... (1)

Now,

⇒ OB = OC

⇒ OB² = OC²

⇒ (x - 0)² + (y - 4)² = (x - 4)² + (y - 6)²

⇒ x² + y² - 8y + 16 = x² - 8x + 16 + y² - 12y + 36

⇒ 8x + 4y = 36

⇒ 2x + y = 9 .... - (2)

On solving (1) and (2), we get

⇒ x = 3 and y = 3

∴ (3, 3) is the centre of the circle.

We know radius is the distance between the centre and any point on the circle.

Let ‘r’ be the radius of the circle.

⇒ r = √10

∴ The radius of the circle is √10.