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
• B. 3- 2√3
• C. 4√5
• D. 3+2√5

Answer 1

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= $\frac{2}{3}$(height of the triangle xyz)

Height=($\frac{\sqrt{3}}{2}$)(2R) =$\sqrt{3}$R

Therefore R+r =$\frac{2}{3}$($\sqrt{3}$R) $\Rightarrow$$\frac{r}{R}$= $\frac{(2-\sqrt{3})}{\sqrt{3}}$