Consider a series of n concentric circles C₁, C₂, , C_n with radii r₁, r₂, r₃, , r_n respectively satisfying r₁r₂r₃ .r_n and r₁=10 The circles are such that the chord of contact of tangents from any point on C_i to C_ i+1 is a tangent t

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Consider a series of n concentric circles C₁, C₂, , C_n with radii r₁, r₂, r₃, , r_n respectively satisfying r₁>r₂>r₃> .>r_n and r₁=10 The circles are such that the chord of contact of tangents from any point on C_i to C_ i+1 is a tangent to C_ i+2 where i=1,2,3, . Find the value of _ n _ r=1 ^n r_i , if the angle between the tangents from any point of C₁ to C₂ is 60^ .

Quantrex Academy · Accepted Answer

Correct answer: 20

x^2+y^2=r₁^2 Chord of contact : x x₁+y y₁=r₂^2 aligned & p=r & | r₂^2 x₁^2+y₁^2 |=r₃ | r₂^2 r₁ |=r₃ & r₂^2=r₁ r₃ aligned Hence, r₁, r₂, r₃, . G.P. 60^ = r₂ 10 = 1 2 r₂=5 Hence, G.P. is 10,5, 5 2 , . _ n _ r=1 ^n r_i= 10 1- 1 2 =20

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