The equation of the biggest circle is

x 2 + y 2 = 1 2

Clearly, it is centred at O (0, 0) and has radius 1.

Let the radii of the other two circles be 1 – r, 1 – 2r, where r > 0.

Thus, the equations of the concentric circles are :

x 2 + y 2 = 1 … (1)

x 2 + y 2 = (1 – r) 2 … (2)

x 2 + y 2 = (1 – 2r) 2 … (3)

Clearly, y = x + 1 cuts the circle (1) at (1, 0) and (0, 1). This line will cut circles (2) and (3) in real and distinct points if

< 1 – r and < 1 – 2r

⇒ < 1 – r and < 1 – 2r

⇒ r < 1 – and r <

⇒ r <

⇒ r ∈ [  r > 0]

Hence is the correct answer.