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Question Number 38613 by rish@bh last updated on 27/Jun/18
The radius of the largest circle which  passes through (1,2) and (3,4) and lies  completely in the first quadrant is  A) 3  B) 2  C) (√6)  D) 2(√5)  I got the answer as 2 but the answer   given is 2(√5).
Theradiusofthelargestcirclewhichpassesthrough(1,2)and(3,4)andliescompletelyinthefirstquadrantisA)3B)2C)6D)25Igottheansweras2buttheanswergivenis25.
Answered by ajfour last updated on 27/Jun/18
The larger circle′s major segment  lies below the line joining the  two points.The circle then touches  the x-axis. Therefore let  the eq. of ⊥ bisector of the  chord from (1,2) to (3,4) is    y−3=2−x   ⇒    x+y =5  let center of circle be  (h,r)  where r is its radius.   centre lies on x+y=5   ⇒     h+r =5  Further      (h−1)^2 +(r−2)^2 =r^2   ⇒  (4−r)^2 +(r−2)^2 = r^2   or     r^2 −12r+20 =0          (r−6)^2 =16   ⇒  r = 6±4            r=10, 2  as  h > 0  ⇒   r = 2 .
Thelargercirclesmajorsegmentliesbelowthelinejoiningthetwopoints.Thecirclethentouchesthexaxis.Thereforelettheeq.ofbisectorofthechordfrom(1,2)to(3,4)isy3=2xx+y=5letcenterofcirclebe(h,r)whererisitsradius.centreliesonx+y=5h+r=5Further(h1)2+(r2)2=r2(4r)2+(r2)2=r2orr212r+20=0(r6)2=16r=6±4r=10,2ash>0r=2.
Commented by rish@bh last updated on 27/Jun/18
Thank you
Thankyou
Commented by prakash jain last updated on 28/Jun/18
if circle touches y−axis  (h−1)^2 +(r−2)^2 =h^2   (h−1)^2 +(3−h)^2 =h^2   h^2 −8h+10=0  h=((8±(√(24)))/2)=4±2(√3)  r=1∓2(√3)  r=1+2(√3),h=4−2(√3)  possible center for touch y−axis  (4+2(√3),1−2(√3))  (4−2(√3),1+2(√3))  radius=4−2(√3)
ifcircletouchesyaxis(h1)2+(r2)2=h2(h1)2+(3h)2=h2h28h+10=0h=8±242=4±23r=123r=1+23,h=423possiblecenterfortouchyaxis(4+23,123)(423,1+23)radius=423

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