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Question
P(5 , -8) , Q (2 , -9) and R(2 , 1) are the vertices of a triangle. Find tyhe circumcentre and the circumradius of the triangle.
Solution
Circumcircle of Δ PQR will pass through its vertices P , Q and R.
OP = OQ
⇒ OP2 = OQ2
(x - 5)2 + (y + 8)2 = (x - 2)2 + (y + 9)2
⇒ 25 - 10x + 64 + 16y = 4 - 4x + 81 + 18 y
c - 6x - 2y + 4 =0
OQ = OR ...(radii of square circle)
OQ2 = OR2
(x - 2)2 + (y + 9)2 = (x - 2)2 + (y - 1)2
⇒ 81 + 18 y = 1 - 2y
⇒ 20 y = - 80
y = -4 ......(2)
-6x + 8+4 = 0 ......[from (2)]
⇒ -6x = -12
⇒ x = 2
Circumcentre of Δ PQR is O (2 , -4)
Circumcentre = `sqrt ((2 - 5)^2 + (-4 + 8)^2)`
`= sqrt (9 + 16) = sqrt 25` = 5 units
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