JEE Advanced 2022 · Paper 2 · Mathematics Q12

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MathematicsMultiple correct+4 / -2Question 12 of 54

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?

n = 4

, then

(\sqrt2 - 1)r < R

n = 5

, then

r < R

n = 8

, then

(\sqrt2 - 1)r < R

n = 12

, then

\sqrt2(\sqrt3 + 1)r > R

Answer and worked solution

Official answer: C, D

Geometry. The centres of $G_1,\dots,G_n$ lie on a circle of radius $R+r$ about the centre of $G$ . Neighbouring centres are $2r$ apart and subtend $\frac{2\pi}n$ , so
$\sin\frac\pi n=\frac r{R+r}\implies R=r\left(\frac1{\sin(\pi/n)}-1\right).$

Check each option.

(A) $n=4$ : $R=(\sqrt2-1)r$ . That is equality, so the strict inequality fails.
(B) $n=5$ : $R=r\left(\frac1{0.588}-1\right)\approx0.70r<r$ . False.
(C) $n=8$ : $R=r\left(\frac1{0.383}-1\right)\approx1.61r>(\sqrt2-1)r$ . True.
(D) $n=12$ : $\frac1{\sin15^\circ}=\sqrt2(\sqrt3+1)$ , so $R=\left(\sqrt2(\sqrt3+1)-1\right)r<\sqrt2(\sqrt3+1)r$ . True.

(C), (D).

Topics:Equation of a circle Trigonometric ratios and identities

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JEE Advanced 2022 · Paper 2, Mathematics question 12