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Three circles of radii a, b, c (a＜b＜c) touch each other externally. If they have x axis as their common yangent, then: 1/ √ a = 1/ √ b + 1/ √ c 1/ √ b = 1/ √ a + 1/ √ c a, b, c are in AP √ a, √ b, √ c are in AP plz explain too

Three circles of radii a, b, c (a＜b＜c) touch each other externally. If they have x axis as their common yangent, then:
  1.   1/√ a = 1/√ b  +  1/√ c
  2. 1/√ b =  1/√ a  +  1/√ c
  3. a, b, c are in AP
  4. √ a, √ b, √ c are in AP
plz explain too

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
3 years ago
dear student there is a theorem called Descartes' circle theorem for 4 kissing circles (you can google it and look up the wikipedia article for more info and refer the SPECIAL CASES section).
using that, we observe that (k1+k2+k3)^2= 2(k1^2+k2^2+k3^2) where k1, k2 and k3 are a, b, c resp
simplifying k1^2+k2^2+k3^2= 2(k1k2+k2k3+k3k1)
k1^2 – 2k1(k2+k3) + (k2 – k3)^2= 0
or k1= k2+k3 ± 2sqrt(k2k3)
k1= [sqrt(k2) ± sqrt(k3)]^2
or sqrt(k1)= sqrt(k2) ± sqrt(k3)
1/sqrt(a)= 1/sqrt(b) ± 1/sqrt(c)
since a＜b＜c, -ve sign is not possible
hence 1/√ a = 1/√ b  +  1/√ c
KINDLY APPROVE :))

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