Three distinct points A, B and C are given in the 2 - dimensional coordinate plane such that the ratio of the distance of any one of them from the point ( 1 , 0 ) to the distance from the point ( − 1

Question

Three distinct points A, B and C are given in the 2 - dimensional coordinate plane such that the ratio of the distance of any one of them from the point ( 1 , 0 ) to the distance from the point ( − 1 , 0 ) is equal to 3 1 ​ . Then the circumcentre of the triangle A BC is at the point

TestHub · Accepted Answer

​ P = ( 1 , 0 ) ; Q ( − 1 , 0 ) Let A = ( x , y ) A Q A P ​ = BQ BP ​ = CQ CP ​ = 3 1 ​ ⇒ 3 A P = A Q ⇒ 9 A P 2 = A Q 2 ⇒ 9 ( x − 1 ) 2 + 9 y 2 = ( x + 1 ) 2 + y 2 ⇒ 9 x 2 − 18 x + 9 + 9 y 2 = x 2 + 2 x + 1 + y 2 ⇒ 8 x 2 − 20 x + 8 y 2 + 8 = 0 ⇒ x 2 + y 2 − 2 5 ​ x + 1 = 0 ​ ∴ A lies on the circle Similarly B, C are also lies on the same circle ∴ Circumcentre of ABC = Centre of Circle ( 1 ) = ( 4 5 ​ , 0 )

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