Let $$a, b$$ be two real numbers such that $$ab < 0$$. If the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a + ib$$ lies on the circle $$|z - 1| = |2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is :

We are given $$ab < 0$$, $$\frac{1+ai}{b+i}$$ has unit modulus, and $$a + ib$$ lies on $$|z - 1| = |2z|$$.

$$\left|\frac{1+ai}{b+i}\right| = 1 \implies |1+ai| = |b+i|$$

$$\sqrt{1 + a^2} = \sqrt{b^2 + 1} \implies a^2 = b^2$$

Since $$ab < 0$$ (opposite signs), we get $$a = -b$$ (i.e., $$b = -a$$).

$$|(a - 1) + ib| = |2a + 2ib|$$

$$(a-1)^2 + b^2 = 4a^2 + 4b^2$$

Substituting $$b = -a$$:

$$(a-1)^2 + a^2 = 4a^2 + 4a^2$$

$$a^2 - 2a + 1 + a^2 = 8a^2$$

$$2a^2 - 2a + 1 = 8a^2$$

$$6a^2 + 2a - 1 = 0$$

$$a = \frac{-2 \pm \sqrt{4 + 24}}{12} = \frac{-2 \pm 2\sqrt{7}}{12} = \frac{-1 \pm \sqrt{7}}{6}$$

Case 1: $$a = \frac{-1 + \sqrt{7}}{6} \approx 0.274$$, $$b = -a \approx -0.274$$

$$[a] = [0.274] = 0$$, so $$\frac{1 + 0}{4(-0.274)} = \frac{1}{-1.097} \approx -0.91$$ (not in options)

Case 2: $$a = \frac{-1 - \sqrt{7}}{6} \approx -0.607$$, $$b = -a \approx 0.607$$

$$[a] = [-0.607] = -1$$, so $$\frac{1 + (-1)}{4(0.607)} = \frac{0}{2.428} = 0$$

A possible value is 0.

The answer is Option A: 0.