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 step by step, we will analyze the given conditions and find the required values.

### Step 1: Understanding the Set S
We are given the set \( S = \{ z: z = x + iy, y \geq 0, |z - z_0| \leq 1 \} \). This means that \( z \) is a complex number where the imaginary part \( y \) is non-negative, and the distance from \( z \) to \( z_0 \) is at most 1.

**Hint:** Remember that the condition \( |z - z_0| \leq 1 \) describes a circle of radius 1 centered at \( z_0 \).

### Step 2: Analyzing the Conditions on \( z_0 \)
We are also 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:
- \( \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \)
- \( \omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \)

The condition \( |z_0| = |z_0 - \omega| = |z_0 - \omega^2| \) implies that \( z_0 \) is equidistant from the origin and the two cube roots of unity.

**Hint:** Use the property of circumcenters in triangles to find \( z_0 \).

### Step 3: Finding \( z_0 \)
The circumcenter of the triangle formed by the points \( 0 \), \( \omega \), and \( \omega^2 \) can be calculated. The centroid of these points is given by:
\[ z_0 = \frac{0 + \omega + \omega^2}{3} \]
Since \( 1 + \omega + \omega^2 = 0 \), we have:
\[ \omega + \omega^2 = -1 \]
Thus,
\[ z_0 = \frac{-1}{3} \]

**Hint:** Remember that the circumcenter is the average of the vertices of the triangle.

### Step 4: Finding the Minimum Value of \( |z| \)
Next, we need to find the minimum value of \( |z| \) for \( z \) in the set \( S \). Since \( |z - z_0| \leq 1 \), we can rewrite this as:
\[ |z + \frac{1}{3}| \leq 1 \]
This describes a circle centered at \( -\frac{1}{3} \) with a radius of 1.

**Hint:** The minimum distance from the origin to this circle will help us find the least value of \( |z| \).

### Step 5: Calculating the Minimum Distance
The center of the circle is at \( -\frac{1}{3} \), and the radius is 1. The distance from the origin to the center is \( \frac{1}{3} \). The minimum distance from the origin to the circle is:
\[ \text{Minimum distance} = \text{Distance to center} - \text{Radius} = \frac{1}{3} - 1 = -\frac{2}{3} \]
Since distance cannot be negative, the minimum value of \( |z| \) is 0.

**Hint:** Consider the geometry of the situation to determine distances.

### Step 6: Finding the Argument of \( \omega - z_0 \)
Now we need to find the argument of \( \omega - z_0 \):
\[ \omega - z_0 = \omega + \frac{1}{3} = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + \frac{1}{3} = -\frac{1}{6} + i\frac{\sqrt{3}}{2} \]

**Hint:** Use the formula for the argument: \( \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) \).

### Step 7: Calculating the Argument
The argument is given by:
\[ \text{arg}(\omega - z_0) = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{6}}\right) \]
This corresponds to an angle in the second quadrant, which can be calculated as:
\[ \text{arg}(\omega - z_0) = \pi - \tan^{-1}\left(\frac{\sqrt{3}}{\frac{1}{6}}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

**Hint:** Ensure you consider the quadrant when determining the angle.

### Final Result
After analyzing all the steps, we conclude:
- The minimum value of \( |z| \) is \( 0 \).
- The argument of \( \omega - z_0 \) is \( \frac{2\pi}{3} \).