If we draw a line joining the centres of the two circles and leading to the intersection point of the two tangents , that can help us arrive at the solution.
Let r be the radius of the smaller circle so distance between centre and the point of intersection of tangent = sqrt(r2 + r2) = √2 r.
And similarly distance between centre of larger circle and the point of intersection of tangent = √ (22 + 22) = 2√2 r
Also this distance can be written as : radius of larger circle + radius of smaller circle + distance between centre of smaller circle to point of intersection of tangents = 2 + r + √2 r.
Hence we have :
2 + r + √2 r = 2√2
==> r(1 + √2) = 2(√2 - 1)
==> r = 2(√2 - 1) / (√2 + 1)
==> r = [ 2(√2 - 1) / (√2 + 1) ] * [ (√2 - 1) / (√2 - 1) ]
= 2(√2 - 1)2 / [ (√2)2 - (1)2 ]
= 2(√2 - 1)2
= 6 - 4√2
Hence 4) should be the correct answer.