Let n ≥ 3 and let C_1,C_2,..,C_n, be circles with radii r_1,r_2,...,r_n, respectively. Assume that C_i and C_{i+1} touch externally for 1 \leq i\leqn - 1. It is also given that the x-axis and the line y = 2\sqrt2x + 10 are tangential to each of the circles. Then r_1,r_2,...,r_n are in

Correct Answer is : a geometric progression with common ratio 2+3

an arithmetic progression with common difference 3+\sqrt2

a geometric progression with common ratio 3+\sqrt2

an arithmetic progression with common difference 2 + \sqrt3

a geometric progression with common ratio 2 + \sqrt3

KVPY SX SB Exam 20141 Question Paper

Question : 84 of 120

Marks: +1, -0

Let n ≥ 3 and let

C_1,C_2,..,C_n

, be circles with radii

r_1,r_2,...,r_n

, respectively. Assume that

C_i

and

C_{i+1}

touch externally for

1 ≤ i≤n - 1

. It is also given that the x-axis and the line

y = 2√2x + 10

are tangential to each of the circles. Then

r_1,r_2,...,r_n

are in

Solution:

\tanθ ={2 \tanθ \/ 2} /{1 -tan ^2θ\/ 2}

2√2= {2 \tanθ \/ 2} /{1 -tan ^2θ\/ 2}

√2 \tan^2 θ\/2 +\tan θ -2 = 0

\tan θ\/2 = {-1±√{1+8}}/{2√2}

={-1±3}/{2√2}=1/√2\or-√2

∴ \tan θ\/2 =1/√2

∆ \OMN \sin θ\/2 =r_2/{ON+r_1+r_2}

⇒ 1/√3=r_2/{√3r_+r_1+r_2}

√3r_1 + r_1 + r_2 = √3r_2

r_1 ( √3 +1) = r_2 (√ 3 -1)

r_2/r_1={√3+1}/{√3-1}

=(√3+1)^2/2=2+√3

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