Two circles with radii a and b touch each other externally. Let c be the radius of circle which touches this circles as well as the common tangent to the circles.Prove that 1/rootc=1/roota+1/rootb.

Asked by rushabhjain.a | 24th Feb, 2019, 12:24: PM

Expert Answer:

In the figure,
the red circle touches the other two circles as well as their common tangent BD.
Let AF = BD = d
BC = x and CD = d - x
d2 = (a + b)2 - (b - a)= 4ba     using Pythagoras in triangle AEF
d = 2√ab  ....(i)
Similarly in triangle AHG,
x2 = (a + c)2 - (a - c)2 = 4ac
x = 2√ac   .....(ii)
In right triangle GEI,
(d - x)2 = (b + c)2 - (b - c)= 4bc
d - x = 2√bc  ....(iii)
From (i), (ii) and (iii)
2√ab = 2√ac + 2√bc
Dividing √abc
we get 
begin mathsize 16px style fraction numerator 1 over denominator square root of straight c end fraction equals fraction numerator 1 over denominator square root of straight b end fraction plus fraction numerator 1 over denominator square root of straight a end fraction end style

Answered by Sneha shidid | 25th Feb, 2019, 09:47: AM