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Q.

Let 𝐺 be a circle of radius 𝑅 > 0. Let 𝐺1, 𝐺2,…, 𝐺𝑛 be 𝑛 circles of equal radius 𝑟 > 0. Suppose each of the 𝑛 circles 𝐺1, 𝐺2,…, 𝐺𝑛 touches the circle 𝐺 externally. Also, for 𝑖 = 1, 2,…, 𝑛 − 1, the circle 𝐺𝑖 touches 𝐺𝑖+1 externally, and 𝐺𝑛 touches 𝐺1 externally. Then, which of the following statements is/are TRUE ?

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a

If 𝑛 = 5, then r < R

b

If 𝑛 = 12, then 2(3+1)r>R

c

If 𝑛 = 4, then (2−1)r<R

d

If 𝑛 = 8, then (2−1)r<R

answer is C, D.

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Detailed Solution

Refer to diagram,
Question Image

sin⁡πn=rR+r⇒Rr+1=cosec⁡πn⇒R=rcosec⁡πn−1
(A) n=4,R=r(2−1)
(B) n=5,R=rcosec⁡π5−1
⇒ R<rcosec⁡π6−1⇒R<r
 (C) n=8,R=rcosec⁡π8−1
⇒ R>rcosec⁡π4−1⇒R>r(2−1)
(D) n=12,R=rcosec⁡π12−1
⇒ R=[2(3+1)−1]r⇒ R<2(3+1)r

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Let be a circle of radius > 0. Let 1, 2,…, be circles of equal radius > 0. Suppose each of the circles 1, 2,…, touches the circle externally. Also, for = 1, 2,…, − 1, the circle touches +1 externally, and touches 1 externally. Then, which of the following statements is/are TRUE ?