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Question

Find the co-ordinates of the circumcentre of triangle if P(2,7), Q (-5, 8), R (-6, 1) are the vertices of triangle.

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Solution

Let the coordinates of the circumcentre O be x, y.Distance of circumcentre from the vertices P2, 7, Q-5, 8 and R-6, 1 of the trianglewill be equal, as it is equal to the radius of the circle.

OP = OQOP2 = OQ2x-22 +y-72= x--52 +y-82x-22 +y-72= x+52 +y-82x2+4-4x+y2+49-14y=x2+25+10x+y2+64-16y-4x-14y+53=10x-16y+89-14x+2y=36-7x+y=18 ----1

OP = OROP2 = OR2x-22 +y-72= x--62 +y-12x-22 +y-72= x+62 +y-12x2+4-4x+y2+49-14y=x2+36+12x+y2+1-2y-4x-14y+53=12x-2y+37-16x-12y=-164x+3y=4 ----2

Multiplying equation 1 by 3, we get:-21x+3y=54 ----3Subtracting equation 2 from 3, we get: -21x+3y=54 4x+3y=4 - - - -25x =50 x = -2Substituting the value of x in equation 1, we get:-7×-2 + y = 1814+y = 18 y = 4 Therefore, -2, 4 are the coordinates of the circumcentre of triangle PQR.

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