Among the complex numbers z satisfying the condition `| z + 1 - i| le 1 ` then number having the least positive argument is

Options

A. 1-i
B. `-1+i`
C. `-i`
D. i

Answer

To solve the problem, we need to analyze the condition given for the complex number \( z \) and find the one with the least positive argument.

### Step 1: Understand the condition
The condition given is:
\[ | z + 1 - i | \leq 1 \]
This represents the set of complex numbers \( z \) that lie within or on the boundary of a circle in the complex plane.

### Step 2: Rewrite the condition
We can rewrite the condition as:
\[ | z - (-1 + i) | \leq 1 \]
This indicates that the center of the circle is at the point \( (-1, 1) \) in the complex plane.

### Step 3: Identify the radius
The radius of the circle is \( 1 \). Thus, the circle includes all points \( z \) such that the distance from \( z \) to the center \( (-1, 1) \) is less than or equal to \( 1 \).

### Step 4: Determine the boundary points
The boundary of the circle can be expressed as:
\[ z = -1 + 1e^{i\theta} \quad \text{for } \theta \in [0, 2\pi] \]
This represents all points on the circumference of the circle.

### Step 5: Find the points on the boundary
The points on the boundary can be expressed as:
\[ z = -1 + \cos(\theta) + i(1 + \sin(\theta)) \]
where \( \theta \) varies from \( 0 \) to \( 2\pi \).

### Step 6: Calculate the argument
The argument of a complex number \( z = x + iy \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
We need to find the value of \( \theta \) that gives the least positive argument.

### Step 7: Analyze the points
The points on the boundary will have coordinates:
\[ (-1 + \cos(\theta), 1 + \sin(\theta)) \]
To find the least positive argument, we need \( x = -1 + \cos(\theta) \) to be as small as possible while ensuring \( y = 1 + \sin(\theta) \) is positive.

### Step 8: Evaluate specific angles
1. For \( \theta = 0 \):
\[ z = -1 + 1 = 0 + i(1) \quad \text{(arg = } \frac{\pi}{2}\text{)} \]
2. For \( \theta = \frac{\pi}{2} \):
\[ z = -1 + 0 + i(2) \quad \text{(arg = } \frac{3\pi}{2}\text{)} \]
3. For \( \theta = \pi \):
\[ z = -1 - 1 + i(1) = -2 + i(1) \quad \text{(arg = } \tan^{-1}\left(-\frac{1}{2}\right)\text{)} \]
4. For \( \theta = \frac{3\pi}{2} \):
\[ z = -1 + 0 + i(0) \quad \text{(arg = } 0\text{)} \]

### Step 9: Conclusion
After evaluating the angles, the point with the least positive argument occurs at \( z = 0 + i(1) \) which has an argument of \( \frac{\pi}{2} \).

### Final Answer
The complex number \( z \) having the least positive argument is:
\[ z = 0 + i \]