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).Toobtain−theradiusrandthecentreCofthecircle.Solution−Wetaketheequationofthecircleasx2+y2+2gx+2fy+c=0.........(i)whenthecentreisO(−g,−f).....(ii)andtheradiusr=√g2+f2−c........(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=0⟹6g+10f+c=−34........(iv)SimilarlyputtingB(x2,y2)=(−1,1)in(i)weget(−1)2+12+2g×(−1)+2f×1+c=0⟹2g−2f−c=2.......(v).AgainputtingC(x3,y3)=(3,−3)in(i)weget32+(−3)2+2g×3+2f×(−3)+c=0⟹6g−6f+c=−18............(vi)subtracting(vi)from(iv)andsimplifyingwehavef=−1.Adding(vi)from(v)andsimplifyingwehaveg−f=−2.puttingf=−1herewehaveg=−3.Sobyputtingf=−1&g=−3in(i)wehave6×(−3)+10×(−1)+c=−34⟹c=−6.∴ThecentreO(−g,−f)=O(3,1)........(fromi)andtheradiusr=√g2+f2−c.....(from(ii)=√(−3)2+(−1)2−(−6)units=4units.∴ThecentreisO(3,1)&theradiusr=4units.AnsOptionB.