Five circles C1,C2,C3,C4,C5 with radii r1,r2,r3,r4,r5 respectively (r1<r2<r3<r4<r5) be such that Ci and Ci+1 touch each other externally for all i=1,2,3,4. If all the five circles touch two straight lines L1 and L2 and r1=2 and r5=32, then r3 is (units)
A
8
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B
8.0
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C
8.00
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Solution
Let ∠O1AT1=x, then sinx=r2−r1r1+r2
Similarly, sinx=r3−r2r3+r2
Now, r2−r1r1+r2=r3−r2r3+r2 ⇒r2r3+r22−r1r3−r1r2=r1r3−r1r2+r2r3−r22⇒r22=r1r3
So, r1,r2,r3 are in G.P.
Similarly, r1,r2,r3,r4,r5 are in G.P.
Therefore, r23=r1r5 ⇒r3=8 units