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Consider a series of `n` concentric circles C(1),C(2),C(3), ....., C(n) with radii r(1),r(2),r(3), ......., r(n) respectively, satisfying r(1)>r(2)>r(3).... > r(n) & r(1)=10. The circles are such that the chord of contact of tangents from any point on Ci to C(i+1) is a tangent to C(i+2) (i=1,2,3,...). Find the value of r(1)+r(2)+r(3)+.........r(infinity), if the angle between the tangents from any point of C(1) to C(2) is 60.
HiThanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
Hiconsider making the diagram yourselfyou must know that c(n) is the innermost circlei'm assuming tangents form c(i) to c(i+1) of which chordof contact touches c(i+2)try to understand the analysisconsider the triangle made by r(i+1),r(i) and length of tangentangle between length of tangent and r(i)=30(since 60 is given)so sin30={r(i+1) / r(i)}r(i+1)={r(i) / 2}consider one more triangle by r(i+1),half of chord and r(i+2)again cos(60)={r(i+2) /r(i+1)}=>r(i+2)={r(i+1) / 2}=>GP with common ratio 1/2=>sum={a / 1-r}={10 / (1-1/2)}=20Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
HelloThanks & RegardsArun KumarBtech, IIT DelhiAskiitians Faculty
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