Question Number 10455 by amir last updated on 10/Feb/17

Commented bymrW1 last updated on 10/Feb/17

circle C1 with radius r_1  and centre  point M_1 (r_1 ,h_1 )with h_1 =r_1  has two  possibilities:  r_1 = { ((2−(√2))),((2+(√2))) :}    circle C2 with radius r_2  and centre  point M_2 (r_2 ,h_2 )  (h_2 −h_1 )^2 +(r_1 −r_2 )^2 =(r_1 +r_2 )^2   ⇒h_2 =h_1 +2(√(r_1 r_2 ))=r_1 +2(√(r_1 r_2 ))    equation of circle C2 is  (x−r_2 )^2 +(y−h_2 )^2 =r_2 ^2   (x−r_2 )^2 +(y−r_1 −2(√(r_1 r_2 )))^2 =r_2 ^2   it tangents the curve xy=1  (x−r_2 )^2 +((1/x)−r_1 −2(√(r_1 r_2 )))^2 =r_2 ^2      ...(i)    way 1:  solve x in terms of r_2  (I know this is not easy)  since there is only one solution for x (tangent!)  ⇒solution for r_2   ⇒determine h_2     way 2:  solve r_2  in terms of x (I know this is not easy)  since r_2  must be minimum (tangent!)  ⇒(dr_2 /dx)=0  ⇒solve x  ⇒determine r_2   ⇒determine h_2     with r_2  and h_2  you can determine   r_3  and h_3  for circle C3 in similary way...  etc...    I don′t think there exists a general  formula for circle C_n  (n=2,3,...20,...)

Commented byb.e.h.i.8.3.4.1.7@gmail.com last updated on 13/Feb/17

hello dear mrW1.please draw a   diagram in large scale and post it   here.i will solve this quiestion.   tnx4all.

Commented bymrW1 last updated on 14/Feb/17

Commented bymrW1 last updated on 14/Feb/17

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

Commented bymrW1 last updated on 16/Mar/17

I can′t see how it is considered that  the circles C_2  and so on tangent the  cirve xy=1.

Commented byb.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Mar/17

it is applay to find r_1 .