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Question

Find the co-ordinates of the centre of a circle passing through the points A(3,5) B(-1, 2) and C(3, -3). Also find the radius of circle.

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Solution

Let the coordinates of circumcentre O be x, y.Distance of circumcentre from the vertices A3, 5, B-1, 2 and R3, -3 of the trianglewill be equal as it is equal to the radius of the circle.

OA = OBOA2 = OB2x-32 +y-52= x--12 +y-22x-32 +y-52= x+12 +y-22x2+9-6x+y2+25-10y=x2+1+2x+y2+4-4y-6x+34-10y=2x+5-4y-8x-6y=-29 8x+6y=29 ----1

OB = OCOB2 = OC2x--12 +y-22= x-32 +y--32x+12 +y-22= x-32 +y+32x2+1+2x+y2+4-4y=x2+9-6x+y2+9+6y2x-4y+5=-6x+6y+188x-10y=13 ----2

Subtracting equation 2 from 1, we get: 8x+6y=29 8x -10y=13 - + - 16y =16 y = 1Substituting the value of y in equation 1, we get: 8x + 6×1 = 298x= 23 y =23 8Therefore, 23 8, 1 are the coordinates of the circumcentre of triangle PQR

Now, radius =OA=238-32 +1-52=-182 +-42=164+16=102564=5418 units

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