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