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Grade: 7

                        

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.

6 years ago

Answers : (3)

Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							
Hi
Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
6 years ago
Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							Hi
consider making the diagram yourself
you must know that c(n) is the innermost circle
i'm assuming tangents form c(i) to c(i+1) of which chord
of contact touches c(i+2)
try to understand the analysis
consider the triangle made by r(i+1),r(i) and length of tangent
angle 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)}=20
Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
6 years ago
Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							
Hello
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty
6 years ago
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