Question

Find the coordinates of the circumcentre of a triangle whose vertices are (-3,1),(0,-2) and (1,3)

Answer 1

Let the O = (a, b) is the co-ordinate of circumcircle of given triangle.
Let A = (-3,1) , B = (0,-2) and C = (1,3)
then, AO = BO = CO = radius of circumcircle.
so, AO² = BO² = CO²

use distance formula,
(a + 3)² + (b - 1)² = a² + (b + 2)² = (a - 1)² + (b - 3)²

now, (a + 3)² + (b - 1)² = a² + (b + 2)²
a² + 6a + 9 + b² - 2b + 1 = a² + b² + 4b + 4
6a - 6b + 6 = 0
a - b + 1 = 0 ------(1)

Similarly, a² + (b + 2)² = (a - 1)² + (b - 3)²
a² + b² + 4b + 4 = a² - 2a + 1 + b² - 6b + 9
2a + 10b - 6 = 0
a + 5b - 3 = 0 ------(2)

from equations (1) and (2),
5a + a +5 - 3 = 0
6a + 2 = 0 , a = -1/3
and b = -1/3 + 1 = 2/3

hence, centre of circumcircle = (-1/3, 2/3)

Answer 2

open the photos attached..... Hope it helps you better...