Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is

Options

1. 2+√3 : 1
2. 4+√3 : 1
3. 7+4√3 : 1
4. 4+2√3 : 1

Correct Answer

7+4√3 : 1

Explanation

Let X and Y be the radii of the bigger and smaller circle, respectively. For the bigger circle: X = r + 2r/√3 For the smaller circle: Y = 2r/√3 − r Ratio of radii: X / Y = (r + 2r/√3) / (2r/√3 − r) = (1 + 2/√3) / (2/√3 − 1) = (2 + √3) / (2 − √3) Rationalize the denominator: X / Y = ((2 + √3) × (2 + √3)) / ((2 − √3) × (2 + √3)) = (4 + 3 + 4√3) / (4 − 3) = (7 + 4√3) : 1

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