Inside a semi-circle of radius 1 unit, two circles of radii r 1 and r 2 are drawn, each touching the circumference and the diameter of the semi-circle and also touching each other externally.
Find [max. (r 1 +r 2 )] where [x] denotes integral value nearest to x.
Text Solution
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Sol Let (a, r 1 ) and (b, r 2 ) be coordinates of centres of two circles. Since circles touch semicircle hence
1– r 1 = 
⇒ 1 – a 2 = 2r 1
and 1– r 2 = 
and 1 – b 2 = 2r 2
also the circles touch each other
So r 1 + r 2 = 
⇒ (a – b) 2 = 4r 1 r 2
also r 1 + r 2 = 1 – 
≤ 1 – |a| |c| therefore
max (r 1 + r 2 ) = 1 – |a| |c|
where |a| = |c| ⇒ r 1 = r 2 = |a|
Hence 1 – r 1 2 = 2r 1 ⇒ r 1 = 
–1
Hence max.(r 1 + r 2 ) = 2( 
–1)
So [max. (r 1 + r 2 )] = 1
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