if z is a complex number belonging to the set `S={z:|z-2+i|gesqrt(5)}` and `z_(0)inS` such that `(1)/(|z_(n)-1|)` is maximum then arg `((4-z_(0)-overline(z)_(0))/(z_(0)-overline(z)_(0)+2i))` is

Options

A. (a) `(3pi)/(4)`
B. (b) `(pi)/(4)`
C. (c) `- (pi)/(2)`
D. (d) `(pi)/(2)`

Answer

To solve the problem step-by-step, we need to break down the given conditions and find the required argument.

### Step 1: Understand the set S
The set \( S \) is defined as:
\[ S = \{ z : |z - (2 - i)| \geq \sqrt{5} \} \]
This represents the region outside or on the boundary of a circle centered at \( (2, -1) \) with a radius of \( \sqrt{5} \).

### Step 2: Find the point \( z_0 \) in \( S \)
We need to maximize \( \frac{1}{|z_n - 1|} \). This means we need to minimize \( |z_n - 1| \). The point \( z_0 \) that minimizes this distance while still being in the set \( S \) should be on the boundary of the circle defined by \( |z - (2 - i)| = \sqrt{5} \).

### Step 3: Parametrize \( z_0 \)
Let \( z_0 = x_0 + iy_0 \). The distance from \( z_0 \) to \( 1 \) is:
\[ |z_0 - 1| = |(x_0 - 1) + iy_0| = \sqrt{(x_0 - 1)^2 + y_0^2} \]
To minimize this, we need to find the point on the circle that is closest to \( 1 \).

### Step 4: Find the center and radius of the circle
The center of the circle is \( (2, -1) \) and the radius is \( \sqrt{5} \). The distance from the center to the point \( (1, 0) \) is:
\[ d = \sqrt{(2 - 1)^2 + (-1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Since \( \sqrt{2} < \sqrt{5} \), the point \( (1, 0) \) lies inside the circle.

### Step 5: Find the closest point on the circle
The closest point on the boundary of the circle to \( (1, 0) \) can be found by moving from the center \( (2, -1) \) towards \( (1, 0) \) along the line connecting these two points. The unit direction vector from \( (2, -1) \) to \( (1, 0) \) is:
\[ \left( \frac{1-2}{\sqrt{2}}, \frac{0+1}{\sqrt{2}} \right) = \left( -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]
Thus, moving a distance of \( \sqrt{5} \) in this direction gives:
\[ z_0 = (2, -1) + \sqrt{5} \left( -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) = \left( 2 - \frac{\sqrt{5}}{\sqrt{2}}, -1 + \frac{\sqrt{5}}{\sqrt{2}} \right) \]

### Step 6: Calculate \( z_0 + \overline{z_0} \) and \( z_0 - \overline{z_0} \)
The conjugate \( \overline{z_0} \) is:
\[ \overline{z_0} = \left( 2 - \frac{\sqrt{5}}{\sqrt{2}}, -1 - \frac{\sqrt{5}}{\sqrt{2}} \right) \]
Thus,
\[ z_0 + \overline{z_0} = 2\left(2 - \frac{\sqrt{5}}{\sqrt{2}}\right) = 4 - \sqrt{10} \]
\[ z_0 - \overline{z_0} = 2i\left(\frac{\sqrt{5}}{\sqrt{2}}\right) = i\sqrt{10} \]

### Step 7: Substitute into the argument expression
We need to find:
\[ \arg\left(\frac{4 - z_0 - \overline{z_0}}{z_0 - \overline{z_0} + 2i}\right) = \arg\left(\frac{4 - (4 - \sqrt{10})}{i\sqrt{10} + 2i}\right) = \arg\left(\frac{\sqrt{10}}{i(\sqrt{10} + 2)}\right) \]

### Step 8: Simplify and find the argument
This simplifies to:
\[ \arg\left(\frac{\sqrt{10}}{i(\sqrt{10} + 2)}\right) = \arg(\sqrt{10}) - \arg(i(\sqrt{10} + 2)) = 0 - \frac{\pi}{2} = -\frac{\pi}{2} \]

### Final Answer
The argument is:
\[ \boxed{-\frac{\pi}{2}} \]