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 G 1 externally. Then, which of the following statements is/are TRUE?
D. If n = 12 , then 2 3 + 1 r > R
Plotting the diagram of given condition we have, Now each radius of smaller circle will make π n angle at centre of bigger circle, Now applying the sine rule, we get sin π n sin π 2 = r R + r ⇒ R r + 1 = cosec π n ⇒ R = r cosec π n - 1 Now checking all options we get, (A) n = 4 , R = r 2 - 1 (B) n = 5 , R = r cosec π 5 - 1 ⇒ R < r cosec π 6 - 1 ⇒ R < r (C) n = 8 , R = r cosec π 8 - 1 ⇒ R > r cosec π 4 - 1 ⇒ R > r 2 - 1 (D) n = 12 , R = r cosec π 12 - 1 ⇒ R = 2 3 + 1 - 1 r ⇒    R < 2 3 + 1 r
Related: Mathematics — Circle · All PYQ Banks