Find the centre of a circle passing through the points (6, -6), (3, -7) and (3,3).Also find the radius. (Ans: (3, -2), 5 units)
Ans: OA=OB = OC = radius of the circle where O is the centre of the circle and let O be (x, y)
OA2 = OB2 = OC2
OA2 = (x-6)2 + (y+6)2 = x2 + y2 - 12x + 36 + 12y + 36
OB2 = (x-3)2 + (y+7)2 = x2 + y2 - 6x + 9 + 14y + 49
OC2 = (x-3)2 + (y-3)2 = x2 + y2 - 6x + 9 - 6y + 9
OA2 = OB2
x2 + y2 - 12x + 12y + 72 = x2 + y2 - 6x + 14y + 58
- 12x + 12y + 6x - 14y + 72 - 58 = 0
- 6x - 2y + 14 = 0
- 3x - y + 7 = 0 ...............(1)
x2 + y2 - 6x + 9 + 14y + 49 = x2 + y2 - 6x + 9 - 6y + 9
- 6x + 14y + 58 = -6x - 6y + 18
14y + 6y = 18 - 58
20y = - 40
y = - 2 ...............(2) Substituting we get
- 3x + 2 + 7 = 0
- 3x = - 9 x = 3
(x, y) = (3, -2)
Diameter = 32 + 22 - 6(3) + 18 - 6 (-2)
= 9 + 4 - 18 + 18 + 12
= 13 + 12 = 25
Radius = √ 25 = 5 units