Let the locus of any point P(z) in the argand plane is `arg((z-5i)/(z+5i))=(pi)/(4)`. If O is the origin, then the value of `(max.(OP)+min.(OP))/(2)` is

Options

A. `5sqrt2`
B. `5+(5)/(sqrt2)`
C. `5+5sqrt2`
D. `10-(5)/(sqrt2)`

Answer

To solve the problem, we need to analyze the given equation and find the maximum and minimum distances from the origin to the locus defined by the argument condition.

### Step-by-Step Solution:

1. **Understanding the Locus**:
The equation given is:
\[ \arg\left(\frac{z - 5i}{z + 5i}\right) = \frac{\pi}{4} \]
This represents the locus of points \( z \) in the Argand plane. The expression inside the argument can be interpreted as the angle formed by the line segments from the points \( z \) to \( 5i \) and \( z \) to \( -5i \).

2. **Identifying Points**:
Let \( z_1 = 5i \) (which corresponds to the point \( (0, 5) \)) and \( z_2 = -5i \) (which corresponds to the point \( (0, -5) \)). The locus will be a circular arc that subtends an angle of \( \frac{\pi}{4} \) at the origin.

3. **Finding the Center and Radius**:
The center of the circle that passes through points \( z_1 \) and \( z_2 \) is at the midpoint of these two points, which is at \( (0, 0) \). The distance between \( z_1 \) and \( z_2 \) is:
\[ z_1 z_2 = |5i - (-5i)| = |10i| = 10 \]
The radius \( r \) of the circle can be calculated using the fact that the angle subtended at the center by the endpoints of the arc is \( \frac{\pi}{4} \).

4. **Using the Circle Properties**:
Since the angle at the center is \( \frac{\pi}{4} \), the radius \( r \) can be derived using the relationship between the radius and the chord length:
\[ r^2 + r^2 = (z_1 z_2)^2 \implies 2r^2 = 10^2 \implies 2r^2 = 100 \implies r^2 = 50 \implies r = 5\sqrt{2} \]

5. **Finding Maximum and Minimum Distances**:
- The maximum distance \( OP \) occurs at the farthest point on the circle from the origin, which is:
\[ \text{Max } OP = r + \text{distance from center to origin} = 5\sqrt{2} + 0 = 5\sqrt{2} \]
- The minimum distance \( OP \) occurs at the closest point on the circle to the origin, which is:
\[ \text{Min } OP = r - \text{distance from center to origin} = 5\sqrt{2} - 5 = 5(\sqrt{2} - 1) \]

6. **Calculating the Final Expression**:
Now we need to compute:
\[ \frac{\text{Max } OP + \text{Min } OP}{2} = \frac{5\sqrt{2} + 5(\sqrt{2} - 1)}{2} = \frac{5\sqrt{2} + 5\sqrt{2} - 5}{2} = \frac{10\sqrt{2} - 5}{2} = 5\sqrt{2} - \frac{5}{2} \]

### Final Answer:
\[ \frac{\text{Max } OP + \text{Min } OP}{2} = 5\sqrt{2} - \frac{5}{2} \]