Properties of TriangleHard
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 
. Then the circumcentre of the triangle ABC is at the point
. Then the circumcentre of the triangle ABC is at the pointOptions
A.(0, 0)
B.
C.
D.
Solution
P = (1, 0) ; Q(-1, 0)
Let A = (x, y)
....(1)
⇒ 3AP = AQ ⇒ 9AP2 = AQ2 ⇒ 9(x - 1)2 + 9y2 = (x + 1)2 + y2
⇒ 9x2 - 18x + 9 + 9y2 = x2 + 2x + 1 + y2 ⇒ 8x2 - 20x + 8y2 + 8 = 0
⇒ x2 + y2
x + 1 = 0 ....(1)
∴ A lies on the circle
Similarly B, C are also lies on the same circle
∴ Circumcentre of ABC = Centre of Circle (1) =
Let A = (x, y)
....(1)⇒ 3AP = AQ ⇒ 9AP2 = AQ2 ⇒ 9(x - 1)2 + 9y2 = (x + 1)2 + y2
⇒ 9x2 - 18x + 9 + 9y2 = x2 + 2x + 1 + y2 ⇒ 8x2 - 20x + 8y2 + 8 = 0
⇒ x2 + y2
x + 1 = 0 ....(1) ∴ A lies on the circle
Similarly B, C are also lies on the same circle
∴ Circumcentre of ABC = Centre of Circle (1) =
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