Find the coordinate of the circumcenter …

 
Let A(3, 0), B(-1, -6) and C(4, -1) be the given points.
Let O(x, y) be the circumcentre of the triangle.
OA = OB = OC
OA² = OB²
(x - 3)² + (y - 0)² = (x + 1)² + (y + 6 )²
{tex}\Rightarrow{/tex} x² + 9 - 6x + y² = x² + 1 +2x + y² + 36 + 12y
{tex}\Rightarrow{/tex} x² - 6x + y² - x² - 2x - y² - 12y = 1 + 36 - 9
{tex}\Rightarrow{/tex} -8x - 12y = 28
{tex}\Rightarrow{/tex} -2x - 3y = 7
{tex}\Rightarrow{/tex} 2x + 3y = -7 ........(i)
Again,
OB² = OC²
(x + 1)² + (y + 6)² = (x - 4)² + (y + 1)²
{tex}\Rightarrow{/tex} x² + 1 + 2x + y² + 36 +12y = x² +16 - 8x + y² + 1 + 2y
{tex}\Rightarrow{/tex} x² + 2x + y² + 12y - x² + 8x + y² - 2y = 16 + 1 - 1 - 36
10x + 10y = -20
x + y = -2 ....... (ii)
Solving (i) and (ii), we get
x = 1, y = -3
Hence circumcentre of the triangle is (1, -3)
Circumradius {tex}= \sqrt { ( 1 + 1 ) ^ { 2 } + ( - 3 + 6 ) ^ { 2 } }{/tex}
{tex}= \sqrt { ( 2 ) ^ { 2 } + ( 3 ) ^ { 2 } }{/tex}
{tex}= \sqrt { 4 + 9 }{/tex}
{tex}= \sqrt { 13 }{/tex} units.