If `0 le "arg"(z) le pi/4`, then the least value of `|z-i|`, is

Options

A. 1
B. `1/sqrt(2)`
C. `sqrt(2)`
D. none of these

Explanation

To find the least value of \( |z - i| \) given that \( 0 \leq \arg(z) \leq \frac{\pi}{4} \), we can follow these steps:

### Step 1: Understand the Region of \( z \)

The argument \( \arg(z) \) being between \( 0 \) and \( \frac{\pi}{4} \) means that \( z \) lies in the first quadrant of the complex plane, specifically between the positive real axis and the line \( y = x \). This can be represented as:
\[ z = x + iy \quad \text{where } y \leq x \text{ and } x \geq 0 \]

### Step 2: Express \( |z - i| \)

We need to find the distance from the point \( z \) to the point \( i \) (which is \( 0 + 1i \)). The distance can be expressed as:
\[ |z - i| = |(x + iy) - (0 + 1i)| = |x + i(y - 1)| = \sqrt{x^2 + (y - 1)^2} \]

### Step 3: Use the Condition \( y \leq x \)

Since \( z \) lies below the line \( y = x \), we can substitute \( y \) with \( x \) to minimize the distance. Thus, we can express \( y \) as:
\[ y = kx \quad \text{where } 0 \leq k \leq 1 \]

### Step 4: Substitute \( y \) into the Distance Formula

Now, substituting \( y = kx \) into the distance formula gives:
\[ |z - i| = \sqrt{x^2 + (kx - 1)^2} \]
Expanding this:
\[ = \sqrt{x^2 + (k^2x^2 - 2kx + 1)} = \sqrt{(1 + k^2)x^2 - 2kx + 1} \]

### Step 5: Minimize the Distance

To minimize this expression, we can differentiate with respect to \( x \) and set the derivative to zero. However, we can also find the minimum geometrically. The minimum distance from the point \( (0, 1) \) to the line \( y = x \) occurs at the foot of the perpendicular from \( (0, 1) \) to the line \( y = x \).

### Step 6: Find the Foot of the Perpendicular

The foot of the perpendicular from the point \( (0, 1) \) to the line \( y = x \) can be found by solving:
1. The slope of the line \( y = x \) is \( 1 \).
2. The slope of the perpendicular line is \( -1 \).
3. The equation of the line through \( (0, 1) \) with slope \( -1 \) is:
\[ y - 1 = -1(x - 0) \implies y = -x + 1 \]
4. Setting \( y = x \) equal to \( y = -x + 1 \):
\[ x = -x + 1 \implies 2x = 1 \implies x = \frac{1}{2} \]
Thus, \( y = \frac{1}{2} \).

### Step 7: Calculate the Minimum Distance

Now we can calculate the distance from \( (0, 1) \) to \( \left(\frac{1}{2}, \frac{1}{2}\right) \):
\[ |z - i| = \sqrt{\left(\frac{1}{2} - 0\right)^2 + \left(\frac{1}{2} - 1\right)^2} = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Conclusion

Thus, the least value of \( |z - i| \) is:
\[ \frac{1}{\sqrt{2}} \]