Question

Find the coordinates of circumcentre and radius of circumcircle of ∆ABC if A(7, 1), B(3, 5) and C(2, 0) are given.

Answer 1

Let the circumcentre be P(a, b).

The given points are A(7, 1), B(3, 5) and C(2, 0).

The circumcircle passes through the points A, B and C and the thus,

PA = PB = PC

⇒ PA² = PB² = PC²

PA² = PB²

⇒ (3 - a)² + (5 - b)² = (7 - a)² + (1 - b)²

⇒ 9 + a² - 6a + 25 + b² - 10b = 49 + a² - 14a + 1 + b² - 2b

⇒ a - b = 2    ...(1)

PA² = PC²

⇒ (7 - a)² + (1 - b)² = (2 - a)² + (0 - b)²

⇒ 49 + a² - 14a + 1 + b² - 2b = 4 + a² - 4a + b²

⇒ 5a + b = 23     ...(2)

(1) + (2)

`a = 25/6, b = 13/6`

Radius = PC =

= `sqrt((25/6 - 2)^2 + (13/6 - 0)^2)`

= `sqrt((13/6)^2 + (13/6 )^2)`

= `13/6sqrt2`