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Question

The co-ordinates of the centre of a circle passing through the points A(3, 5). B(- I, I) and C(3, -3) and its centre are

A
ThecentreisO(3,1)&theradiusr=4units.
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B
ThecentreisO(3,1)&theradiusr=4units.
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C
ThecentreisO(3,1)&theradiusr=4units.
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D
ThecentreisO(3,1)&theradiusr=4units.
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Solution

The correct option is B ThecentreisO(3,1)&theradiusr=4units.
AcirclepassesthroughA(x1,y1)=(3,5),B(x2,y2)=(1,1)&C(x3,y3)=(3,3).ToobtaintheradiusrandthecentreCofthecircle.SolutionWetaketheequationofthecircleasx2+y2+2gx+2fy+c=0.........(i)whenthecentreisO(g,f).....(ii)andtheradiusr=g2+f2c........(iii)IfthecirclepassesthroughA(x1,y1),B(x2,y2)&C(x3,y3)thenA,B,&Cwillsatisfytheequation(i).PuttingA(x1,y1)=(3,5)in(i)weget32+52+2g×3+2f×5+c=06g+10f+c=34........(iv)SimilarlyputtingB(x2,y2)=(1,1)in(i)weget(1)2+12+2g×(1)+2f×1+c=02g2fc=2.......(v).AgainputtingC(x3,y3)=(3,3)in(i)weget32+(3)2+2g×3+2f×(3)+c=06g6f+c=18............(vi)subtracting(vi)from(iv)andsimplifyingwehavef=1.Adding(vi)from(v)andsimplifyingwehavegf=2.puttingf=1herewehaveg=3.Sobyputtingf=1&g=3in(i)wehave6×(3)+10×(1)+c=34c=6.ThecentreO(g,f)=O(3,1)........(fromi)andtheradiusr=g2+f2c.....(from(ii)=(3)2+(1)2(6)units=4units.ThecentreisO(3,1)&theradiusr=4units.AnsOptionB.

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