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Question

In the adjoining figure ACE is a right angle there are three circles which just touch each other and AC and EC are the tangents to all the three circles. What is the ratio of radii of the largest circle to that of the smallest circle ?

  1. None of the above
  2. 17:122
  3. 1:(17122)
  4. 12:172


Solution

The correct option is C 1:(17122)

In ΔCIE,
CI2=CE2+EI2
But CE = EI
   CI2=2EI2CI=2 EI=2r3CI=2r1+r1+2r2+r3=(2+1)r1+2r2+r3(2+1)r1+2r2+r3=2r3(2+1)r1+2r2=(21)r3     (i)Consider, ΔCHD,(2+1)r1=(21)r2r2=(2+121)r1
Putting the value of r2 in i), we get
(2+1)r1+2{2+121}r1=(21)r3(3+22)r1=(322)r3r3r1=3+22322=(3+22)2=17+122r3r1=(117122)

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