Key Concepts and Formulas

• Complex Number Representation: $z = x + iy$, where $x = \text{Re}(z)$ and $y = \text{Im}(z)$. Also, $z + \bar{z} = 2x$ and $z - \bar{z} = 2iy$.
• Modulus of a Complex Number: $|z| = \sqrt{x^2 + y^2}$, representing the distance from the origin.
• Geometric Interpretation of $|z - z_0| = r$: Circle centered at $z_0$ with radius $r$.

Step-by-Step Solution

Step 1: Convert the first condition to Cartesian form

We are given $|z - 1| = 1$. Substituting $z = x + iy$ allows us to express this in terms of real variables. This is a standard technique for dealing with moduli.

$|x + iy - 1| = 1$ $|(x - 1) + iy| = 1$ $\sqrt{(x - 1)^2 + y^2} = 1$ $(x - 1)^2 + y^2 = 1$

This is the equation of a circle centered at $(1, 0)$ with radius $1$.

Step 2: Convert the second condition to Cartesian form

We are given $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2}$. We use the identities $z + \bar{z} = 2x$ and $z - \bar{z} = 2iy$ to simplify this expression.

$(\sqrt{2} - 1)(2x) - i(2iy) = 2\sqrt{2}$ $2(\sqrt{2} - 1)x - 2i^2y = 2\sqrt{2}$ $2(\sqrt{2} - 1)x + 2y = 2\sqrt{2}$ $(\sqrt{2} - 1)x + y = \sqrt{2}$

This is the equation of a line.

Step 3: Solve for y in terms of x from the line equation

We isolate $y$ in the equation $(\sqrt{2} - 1)x + y = \sqrt{2}$. This will allow us to substitute into the circle equation.

$y = \sqrt{2} - (\sqrt{2} - 1)x$

Step 4: Substitute the expression for y into the circle equation

Substitute $y = \sqrt{2} - (\sqrt{2} - 1)x$ into $(x - 1)^2 + y^2 = 1$. This will give us a quadratic in $x$.

$(x - 1)^2 + (\sqrt{2} - (\sqrt{2} - 1)x)^2 = 1$ $x^2 - 2x + 1 + ( \sqrt{2} - (\sqrt{2} - 1)x)^2 = 1$ $x^2 - 2x + 1 + 2 - 2\sqrt{2}(\sqrt{2}-1)x + (\sqrt{2}-1)^2x^2 = 1$ $x^2 - 2x + 1 + 2 - (4 - 2\sqrt{2})x + (2 - 2\sqrt{2} + 1)x^2 = 1$ $x^2 - 2x + 1 + 2 - 4x + 2\sqrt{2}x + (3 - 2\sqrt{2})x^2 = 1$ $x^2 - 2x + 2 - 4x + 2\sqrt{2}x + 3x^2 - 2\sqrt{2}x^2 = 0$ $(4 - 2\sqrt{2})x^2 + (-6 + 2\sqrt{2})x + 2 = 0$ $(2 - \sqrt{2})x^2 + (\sqrt{2} - 3)x + 1 = 0$

Step 5: Solve the quadratic equation for x

We have the quadratic equation $(2 - \sqrt{2})x^2 + (\sqrt{2} - 3)x + 1 = 0$. We can use the quadratic formula or try to factor. Trying $x = 1$:

$(2 - \sqrt{2}) - 3 + \sqrt{2} + 1 = 0$ $3 - 3 = 0$

So, $x = 1$ is a root. Let the roots be $x_1$ and $x_2$. Since the product of roots is $\frac{c}{a}$:

$x_1 x_2 = \frac{1}{2 - \sqrt{2}} = \frac{2 + \sqrt{2}}{4 - 2} = \frac{2 + \sqrt{2}}{2} = 1 + \frac{\sqrt{2}}{2}$

Since $x_1 = 1$, we have $x_2 = 1 + \frac{\sqrt{2}}{2}$.

Step 6: Find the corresponding y values

• For $x = 1$: $y = \sqrt{2} - (\sqrt{2} - 1)(1) = \sqrt{2} - \sqrt{2} + 1 = 1$. So $z_A = 1 + i$.
• For $x = 1 + \frac{\sqrt{2}}{2}$: $y = \sqrt{2} - (\sqrt{2} - 1)(1 + \frac{\sqrt{2}}{2}) = \sqrt{2} - (\sqrt{2} + 1 - 1 - \frac{\sqrt{2}}{2}) = \sqrt{2} - (\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2}$. So $z_B = 1 + \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}$.

Step 7: Find the magnitudes of the complex numbers

• $|z_A| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$
• $|z_B| = |1 + \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}| = \sqrt{(1 + \frac{\sqrt{2}}{2})^2 + (\frac{\sqrt{2}}{2})^2} = \sqrt{1 + \sqrt{2} + \frac{1}{2} + \frac{1}{2}} = \sqrt{2 + \sqrt{2}}$

Since $\sqrt{2 + \sqrt{2}} > \sqrt{2}$, we have $z_1 = z_B = 1 + \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}$ and $z_2 = z_A = 1 + i$.

Step 8: Calculate $\sqrt{2}z_1 - z_2$

$\sqrt{2}z_1 = \sqrt{2}(1 + \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) = \sqrt{2} + 1 + i$ $\sqrt{2}z_1 - z_2 = \sqrt{2} + 1 + i - (1 + i) = \sqrt{2}$

Step 9: Calculate $|\sqrt{2}z_1 - z_2|^2$

$|\sqrt{2}z_1 - z_2|^2 = |\sqrt{2}|^2 = 2$

Common Mistakes & Tips

• Be careful with algebraic manipulations, especially when expanding squared terms.
• When comparing magnitudes, compare the squared magnitudes to avoid square roots.
• Recognize the geometric interpretations of the equations to help visualize the problem.

Summary

We converted the given complex number equations into Cartesian form, solved the resulting system of equations to find the complex numbers in the set S, and then used the modulus definition to identify $z_1$ and $z_2$. Finally, we calculated the value of the expression $|\sqrt{2}z_1 - z_2|^2$, which simplifies to $2$.

Final Answer The final answer is \boxed{2}, which corresponds to option (D).