The number of complex numbers z, satisfying $$|z|=1\text{ and }|\frac{z}{\overline{z}}+\frac{\overline{z}}{z}| = 1$$, is :

We need to find the number of complex numbers z satisfying $$|z| = 1$$ and $$\left|\frac{z}{\bar{z}} + \frac{\bar{z}}{z}\right| = 1$$.

Let $$z = e^{i\theta}$$

Since $$|z| = 1$$, we have $$\bar{z} = e^{-i\theta}$$.

$$\frac{z}{\bar{z}} = e^{2i\theta}$$ and $$\frac{\bar{z}}{z} = e^{-2i\theta}$$

$$\frac{z}{\bar{z}} + \frac{\bar{z}}{z} = e^{2i\theta} + e^{-2i\theta} = 2\cos 2\theta$$

Apply the condition

$$|2\cos 2\theta| = 1$$

$$\cos 2\theta = \pm \frac{1}{2}$$

Solve for $$\theta$$

Case 1: $$\cos 2\theta = \frac{1}{2}$$

$$2\theta = \pm \frac{\pi}{3} + 2n\pi$$

$$\theta = \pm \frac{\pi}{6} + n\pi$$

In $$[0, 2\pi)$$: $$\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$ — 4 values

Case 2: $$\cos 2\theta = -\frac{1}{2}$$

$$2\theta = \pm \frac{2\pi}{3} + 2n\pi$$

$$\theta = \pm \frac{\pi}{3} + n\pi$$

In $$[0, 2\pi)$$: $$\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$$ — 4 values

Total count

Total = 4 + 4 = 8 complex numbers.

The correct answer is Option 2: 8.