Let $$ z_1,z_2 \text{ and } z_3$$ be three complex numbers on the circle $$ \mid z \mid = 1 $$ with $$ arg(z_1)=\frac{-\pi}{4},arg(z_2)=0 \text{ and } arg(z_3)=\frac{\pi}{4}$$.  If $$\mid z_1\overline{z}_2+z_2\overline{z}_3+z_3\overline{z}_1 \mid^{2}= \alpha+ \beta \sqrt{2}, \alpha, \beta \in Z$$, then the value of $$ \alpha^{2}+\beta^{2} \text{ is :} $$

Given three complex numbers on the unit circle $$|z| = 1$$ with arguments $$\arg(z_1) = -\frac{\pi}{4}$$, $$\arg(z_2) = 0$$, and $$\arg(z_3) = \frac{\pi}{4}$$.

Since they lie on the unit circle, they can be expressed in exponential form as:

$$z_1 = e^{-i\pi/4}, \quad z_2 = e^{i \cdot 0} = 1, \quad z_3 = e^{i\pi/4}$$

The conjugates are:

$$\overline{z_1} = e^{i\pi/4}, \quad \overline{z_2} = e^{-i \cdot 0} = 1, \quad \overline{z_3} = e^{-i\pi/4}$$

We need to compute the expression:

$$S = z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}$$

Substituting the values:

$$z_1 \overline{z_2} = e^{-i\pi/4} \cdot 1 = e^{-i\pi/4}$$

$$z_2 \overline{z_3} = 1 \cdot e^{-i\pi/4} = e^{-i\pi/4}$$

$$z_3 \overline{z_1} = e^{i\pi/4} \cdot e^{i\pi/4} = e^{i(\pi/4 + \pi/4)} = e^{i\pi/2} = i$$

Adding these together:

$$S = e^{-i\pi/4} + e^{-i\pi/4} + i = 2e^{-i\pi/4} + i$$

Now, $$e^{-i\pi/4} = \cos(-\pi/4) + i \sin(-\pi/4) = \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} (1 - i)$$. Therefore:

$$2e^{-i\pi/4} = 2 \cdot \frac{1}{\sqrt{2}} (1 - i) = \sqrt{2} (1 - i) = \sqrt{2} - i\sqrt{2}$$

So,

$$S = (\sqrt{2} - i\sqrt{2}) + i = \sqrt{2} + i(1 - \sqrt{2})$$

To find $$|S|^2$$, recall that for a complex number $$z = x + iy$$, $$|z|^2 = x^2 + y^2$$. Here, $$x = \sqrt{2}$$ and $$y = 1 - \sqrt{2}$$:

$$|S|^2 = (\sqrt{2})^2 + (1 - \sqrt{2})^2 = 2 + \left(1 - 2\sqrt{2} + (\sqrt{2})^2\right) = 2 + \left(1 - 2\sqrt{2} + 2\right) = 2 + (3 - 2\sqrt{2}) = 5 - 2\sqrt{2}$$

The expression $$|S|^2 = \alpha + \beta \sqrt{2}$$ is given, so comparing:

$$5 - 2\sqrt{2} = \alpha + \beta \sqrt{2} \implies \alpha = 5, \quad \beta = -2$$

Now, compute $$\alpha^2 + \beta^2$$:

$$\alpha^2 + \beta^2 = 5^2 + (-2)^2 = 25 + 4 = 29$$

Thus, the value is 29.