Question

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

Answer 1

Let O = (a, b) is the centre of circumcircle .
then, AO = BO = CO = radius of circumcircle
$\because$ AO = BO = CO
$\therefore$ AO² = BO² = CO²

use distance formula,
so, (a - 7)² + (b - 1)² = (a - 3)² + (b - 5)² = (a + 2)² + b²

now, (a - 7)² + (b - 1)² = (a - 3)² + (b - 5)²
a² - 14a + 49 + b² - 2b + 1 = a² - 6a + 9 + b² - 10b + 25
-14a + 6a - 2b + 10b + 50 - 34 = 0
-8a + 8b + 16 = 0
a - b - 2 = 0
a = b + 2 -------(1)

Similarly, (a - 3)² + (b - 5)² = (a - 2)² + b²
a² - 6a + 9 + b² - 10b + 25 = a² - 4a + 4 + b²
-6a + 4a - 10b + 34 - 4 = 0
-2a - 10b + 30 = 0
a + 5b = 15 ------(2)

from equations (1) and (2),
b + 2 + 5b = 15
6b = 13 , b = 13/6
and a = 19/6

hence, centre of circumcircle = (19/6,13/6)
now, radius of circumcircle = $\sqrt{(19/6-7)^2+(13/6-1)^2}$
= $\sqrt{538}/6$