Question:
Let $G$ be a circle of radius $R>0$. Let $G _1, G _2, \ldots, G _{ n }$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G _1, G _2, \ldots, G _{ n }$ touches the circle $G$ externally. Also, for $i =1,2, \ldots, n -1$, 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?
Let $G$ be a circle of radius $R>0$. Let $G _1, G _2, \ldots, G _{ n }$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G _1, G _2, \ldots, G _{ n }$ touches the circle $G$ externally. Also, for $i =1,2, \ldots, n -1$, 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?
Updated On: Apr 23, 2024
If \(n =4\), then \((\sqrt{2}-1) r < R\)
If \(n =5\), then \(r < R\)
If \(n =8\), then \((\sqrt{2}-1) r < R\)
If \(n =12\), then \(\sqrt{2}(\sqrt{3}+1) r > R\)
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The Correct Option is C, D
Solution and Explanation
So, the correct option is
(C) If \(n =8\), then \((\sqrt{2}-1) r < R\)
(D) If \(n =12\), then \(\sqrt{2}(\sqrt{3}+1) r > R\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.