Find the circum - centre and circum - radius of the triangle whose vertices are (- 2, 3), (2, - 1) and (4, 0).

Given that we need to find the circum - centre and circum - radius of the triangle whose vertices are A(- 2, 3), B(2, - 1), C(4, 0).

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

We know that distance from circum - centre to any vertex 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 - (- 2))² + (y - 3)² = (x - 2)² + (y - (- 1))²

⇒ (x + 2)² + (y - 3)² = (x - 2)² + (y + 1)²

⇒ x² + 4x + 4 + y² - 6y + 9 = x² - 4x + 4 + y² + 2y + 1

⇒ 8x - 8y = - 8

⇒ x - y = - 1 ..... (1)

Now,

⇒ OB = OC

⇒ OB² = OC²

⇒ (x - 2)² + (y - (- 1))² = (x - 4)² + (y - 0)²

⇒ (x - 2)² + (y + 1)² = (x - 4)² + (y)²

⇒ x² - 4x + 4 + y² + 2y + 1 = x² - 8x + 16 + y²

⇒ 4x + 2y = 11 .... - (2)

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

⇒ and

∴ 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 circum - radius of the circle.

∴ The radius of the circle is .