Let 𝐺 be a circle of radius 𝑅 > 0. Let 𝐺₁, 𝐺₂,…, 𝐺_𝑛 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 𝐺_𝑖touches 𝐺_𝑖+1 externally, and 𝐺_𝑛 touches 𝐺₁ externally. Then, which of the following statements is/are TRUE ?

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If 𝑛 = 5, then r < R

If 𝑛 = 12, then $\sqrt{2} (\sqrt{3} + 1) r > R$

If 𝑛 = 4, then $(\sqrt{2} - 1) r < R$

If 𝑛 = 8, then $(\sqrt{2} - 1) r < R$

answer is C, D.

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Detailed Solution

Refer to diagram,
Question Image

$\begin{array}{l} \sin (\frac{π}{n}) = \frac{r}{R + r} \\ \Rightarrow \frac{R}{r} + 1 = cosec (\frac{π}{n}) \\ \Rightarrow R = r [cosec (\frac{π}{n}) - 1] \end{array}$
(A) $n = 4, R = r (\sqrt{2} - 1)$
(B) $n = 5, R = r (cosec \frac{π}{5} - 1)$
$\Rightarrow R < r (cosec \frac{π}{6} - 1) \Rightarrow R < r$
$(C) n = 8, R = r [cosec (\frac{π}{8}) - 1]$
$\begin{array}{l} \Rightarrow R > r (cosec (\frac{π}{4}) - 1) \\ \Rightarrow R > r (\sqrt{2} - 1) \end{array}$
(D) $n = 12, R = r [cosec (\frac{π}{12}) - 1]$
$\begin{array}{l} \Rightarrow R = [\sqrt{2} (\sqrt{3} + 1) - 1] r \\ \Rightarrow R < \sqrt{2} (\sqrt{3} + 1) r \end{array}$

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