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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.
Solution
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`
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