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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
3.5
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D
4√5
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E
3+2√5
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Solution

The correct option is D (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= 2/3(height of the triangle xyz)

Height=(√3/2)(2R) =√3R

Therefore R+r = 2/3(√3R)  =>r/R= (2-√3)/√3

 

Shortcut:- Using the approximation technique used in class, the radius of the bigger circle: smaller circle is close to 0.2.


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