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Question

# Three balls of equal radius are placed such that they are touching each other. A fourth smaller ball is kept such that it touches the other three. Find the ratio of the radii of smaller to larger ball.

A
(2-√3)/ √3
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B
3- 2√3
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C
4√5
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D
3+2√5
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Solution

## The correct option is A (2-√3)/ √3 Option (a) Let the centers of the large balls be x, y, z and radius R. O is the centre of the smaller ball and radius r... x, y, z form an equilateral triangle with side equal to 2R. O is the centroid of this triangle. Therefore ox=oy=oz=R+r= 23(height of the triangle xyz) Height=(√32)(2R) =√3R Therefore R+r =23(√3R) ⇒rR= (2−√3)√3

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