Let G be a circle of radius R > 0 . Let G 1 , G 2 , … , G n be n circles of equal radius r > 0 . Suppose each of the n circles G 1 , G 2 , … , G n touches the circle G externally. Also, for i = 1 , 2 , … , n - 1 , the circle G i touches G i + 1 externally, and G n touches…

Question

Let G be a circle of radius R &#62; 0 . Let G 1 , G 2 , &#8230; , G n be n circles of equal radius r &#62; 0 . Suppose each of the n circles G 1 , G 2 , &#8230; , G n touches the circle G externally. Also, for i = 1 , 2 , &#8230; , n - 1 , the circle G i touches G i + 1 externally, and G n touches G 1 externally. Then, which of the following statements is/are TRUE?

Aicharya · Accepted Answer

Plotting the diagram of given condition we have,

Now each radius of smaller circle will make $\frac{π}{n}$ angle at centre of bigger circle,

Now applying the sine rule, we get

$\frac{\sin (\frac{π}{n})}{\sin \frac{π}{2}} = \frac{r}{R + r}$

$\Rightarrow \frac{R}{r} + 1 = cosec (\frac{π}{n})$

$\Rightarrow R = r [cosec (\frac{π}{n})] [- 1]$

Now checking all options we get,

(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) $n = 8, R = r [cosec (\frac{π}{8}) - 1]$

$\Rightarrow R > r (cosec (\frac{π}{4}) - 1)$

$\Rightarrow R > r (\sqrt{2} - 1)$

(D) $n = 12, R = r [cosec (\frac{π}{12}) - 1]$

$\Rightarrow R = [\sqrt{2} (\sqrt{3} + 1) - 1] r$

$\Rightarrow R < \sqrt{2} (\sqrt{3} + 1) r$

Let G be a circle of radius R > 0 . Let G 1 , G 2 , … , G n be n circles of equal radius r > 0…

Solution

Let G be a circle of radius R &#62; 0 . Let G 1 , G 2 , &#8230; , G n be n circles of equal radius r &#62; 0…

Solution

Let G be a circle of radius R > 0 . Let G 1 , G 2 , … , G n be n circles of equal radius r > 0…