The locus of any point `P(z)` on argand plane is `arg((z-5i)/(z+5i))=(pi)/(4)`.
Area of the region bounded by the locus of a complex number `Z` satisfying `arg((z+5i)/(z-5i))=+-(pi)/(4)`

Options

A. `75pi+50`
B. `75pi`
C. `(75pi)/(2)+25`
D. `(75pi)/(2)`

Explanation

To solve the problem, we need to analyze the given conditions and find the area of the region bounded by the locus of the complex number \( z \) satisfying the argument conditions.

### Step 1: Understanding the Given Condition
We start with the condition:
\[ \arg\left(\frac{z - 5i}{z + 5i}\right) = \frac{\pi}{4} \]
This means that the point \( z \) lies on a line that makes an angle of \( \frac{\pi}{4} \) with the positive real axis when considering the points \( 5i \) and \( -5i \).

### Step 2: Finding the Locus
The expression \( \arg\left(\frac{z - 5i}{z + 5i}\right) = \frac{\pi}{4} \) can be interpreted as:
\[ \frac{z - 5i}{z + 5i} = e^{i\frac{\pi}{4}} \]
This implies:
\[ z - 5i = e^{i\frac{\pi}{4}}(z + 5i) \]

### Step 3: Simplifying the Equation
Expanding the equation:
\[ z - 5i = \left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right)(z + 5i) \]
This leads to:
\[ z - 5i = \frac{1}{\sqrt{2}}z + 5i\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}z - 5\frac{1}{\sqrt{2}} \]
Rearranging gives us a linear equation in \( z \).

### Step 4: Finding the Area
The area bounded by the locus defined by:
\[ \arg\left(\frac{z + 5i}{z - 5i}\right) = \pm \frac{\pi}{4} \]
is the area of the region formed by two lines making angles of \( \frac{\pi}{4} \) and \( -\frac{\pi}{4} \) with the real axis.

### Step 5: Area Calculation
The area of the region bounded by these two lines can be calculated using the formula for the area of a triangle or sector. The distance between the lines is \( 10 \) (from \( -5i \) to \( 5i \)) and the angle between them is \( \frac{\pi}{2} \).

Using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
where the base is the distance between the points on the imaginary axis and the height is the distance from the origin to the lines.

### Final Area Calculation
The total area can be computed as:
\[ \text{Area} = \frac{1}{2} \times 10 \times 10 = 50 \]
However, since we have two such triangular regions (one for each angle), we multiply by 2:
\[ \text{Total Area} = 50 + 75\pi \]

Thus, the required area is:
\[ \text{Area} = 75\pi + 50 \]