Let `S = { z: x = x+ iy, y ge 0,|z-z_(0)| le 1}`, where `|z_(0)|= |z_(0) - omega|= |z_(0) - omega^(2)|, omega and omega^(2)` are non-real cube roots of unity. Then

Options

A. `z_(0) = - 1`
B. `z_(0) = -1//2`
C. if `z in S`, then least value of |z| is 1
D. `|arg(omega -z_(0)) |=pi//3`

Explanation

To solve the problem, we will analyze the given set \( S \) and the conditions provided.

### Step 1: Understand the given set \( S \)

The set \( S \) is defined as:
\[ S = \{ z : z = x + iy, y \geq 0, |z - z_0| \leq 1 \} \]
This means \( z \) is a complex number where the imaginary part \( y \) is non-negative, and the distance from \( z_0 \) is at most 1.

### Step 2: Analyze the conditions on \( z_0 \)

We are given that:
\[ |z_0| = |z_0 - \omega| = |z_0 - \omega^2| \]
where \( \omega \) and \( \omega^2 \) are the non-real cube roots of unity. The cube roots of unity are:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \quad \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]

### Step 3: Geometric interpretation

The condition \( |z_0| = |z_0 - \omega| = |z_0 - \omega^2| \) implies that \( z_0 \) is equidistant from the origin (0) and the points \( \omega \) and \( \omega^2 \). This means \( z_0 \) is the circumcenter of the triangle formed by the points \( 0, \omega, \) and \( \omega^2 \).

### Step 4: Find the circumcenter

The circumcenter of a triangle formed by the points \( 0, \omega, \) and \( \omega^2 \) can be calculated. The centroid of these points is given by:
\[ C = \frac{0 + \omega + \omega^2}{3} = \frac{\omega + \omega^2}{3} \]
Since \( \omega + \omega^2 = -1 \) (the sum of the roots of unity), we have:
\[ C = \frac{-1}{3} \]
However, the circumcenter lies on the perpendicular bisector of the segment connecting \( \omega \) and \( \omega^2 \). The midpoint of \( \omega \) and \( \omega^2 \) is:
\[ M = \frac{\omega + \omega^2}{2} = \frac{-1}{2} \]
The circumradius \( R \) can be calculated as the distance from the circumcenter to any vertex, which is the same for all vertices.

### Step 5: Determine \( z_0 \)

Given the symmetry and the distances, we find that the circumcenter \( z_0 \) is at:
\[ z_0 = 0 \]
This is incorrect; the circumcenter should be at \( z_0 = 1 \) based on the conditions given in the problem.

### Step 6: Determine the set \( S \)

Now substituting \( z_0 = 1 \) into the set definition:
\[ |z - 1| \leq 1 \]
This describes a disk centered at \( 1 \) with radius \( 1 \), but since \( y \geq 0 \), we only consider the upper half of this disk.

### Step 7: Find the minimum value of \( |z| \)

The minimum value of \( |z| \) occurs at the point on the boundary of the disk closest to the origin. The closest point on the boundary of the disk to the origin is at \( z = 0 \).

### Step 8: Find the argument

The argument of \( \omega - z_0 \) can be calculated:
\[ \text{arg}(\omega - 1) = \text{arg}\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2} - 1\right) = \text{arg}\left(-\frac{3}{2} + i\frac{\sqrt{3}}{2}\right) \]
The angle made with the real axis can be computed as:
\[ \theta = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{3}{2}}\right) = \tan^{-1}\left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6} \]
However, since we are looking for the positive angle, we add \( \pi \):
\[ \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]

### Final Answer

Thus, the final values are:
- \( z_0 = 1 \)
- Minimum value of \( |z| = 0 \)
- Argument \( \text{arg}(\omega - z_0) = \frac{5\pi}{6} \)