inside the circles `x^2+y^2=1` there are three circles of equal radius `a` tangent to each other and to `s` the value of `a` equals to

Options

A. `sqrt(2) (sqrt(2)-1)`
B. `sqrt(3)(2-sqrt(3))`
C. `sqrt(2)(2-sqrt(3))`
D. `sqrt(3)(sqrt(3)-1)`

Answer

B

Explanation

We have `AC = 2a`
and `OA = OC =1 -a` (since circles are touching internally)
Since `DeltaABC`is equilateral, we have
`:. /_AOC = 120^(@)`
Using cosine rule, we have
`cos 120^(@) = ((1-a)^(2)+(1-a)^(2)-(2a)^(2))/(2(1-a)(1-a))`
`rArr -(1)/(2) =(-a^(2)-2a+1)/((1-a)^(2))`
`rArr a^(2)-2a +1 =2a^(2) +4a -2`
`rArr a^(2)+6a -3 =0`
`rArr a = (-6+sqrt(48))/(2) = sqrt(3) (2-sqrt(3))`