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 find the locus of the point \( P(z) \) in the Argand plane given by the equation:

\[ \arg\left(\frac{z - 5i}{z + 5i}\right) = \frac{\pi}{4} \]

### Step 1: Rewrite the expression
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. We can rewrite the expression as:

\[ \arg\left(\frac{(x + iy) - 5i}{(x + iy) + 5i}\right) = \arg\left(\frac{x + i(y - 5)}{x + i(y + 5)}\right) \]

### Step 2: Simplify the expression
To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{x + i(y - 5)}{x + i(y + 5)} \cdot \frac{x - i(y + 5)}{x - i(y + 5)} = \frac{(x)(x) + (y - 5)(y + 5)}{x^2 + (y + 5)^2} \]

This gives us:

\[ \frac{x^2 + y^2 - 25}{x^2 + (y + 5)^2} \]

### Step 3: Set the argument equal to \(\frac{\pi}{4}\)
The argument of a complex number \( \frac{a + bi}{c + di} \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \). Therefore, we have:

\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

This implies:

\[ \frac{y - 5}{x} = 1 \quad \text{or} \quad \frac{y + 5}{x} = -1 \]

### Step 4: Solve for \( y \)
From \( \frac{y - 5}{x} = 1 \):

\[ y - 5 = x \implies y = x + 5 \]

From \( \frac{y + 5}{x} = -1 \):

\[ y + 5 = -x \implies y = -x - 5 \]

### Step 5: Find the intersection points
Now we need to find the intersection of these two lines with the circle defined by the equation derived from the locus:

\[ x^2 + y^2 + 10x - 25 = 0 \]

### Step 6: Substitute \( y \)
Substituting \( y = x + 5 \) into the circle equation:

\[ x^2 + (x + 5)^2 + 10x - 25 = 0 \]

This simplifies to:

\[ x^2 + (x^2 + 10x + 25) + 10x - 25 = 0 \]
\[ 2x^2 + 20x = 0 \implies 2x(x + 10) = 0 \]

Thus, \( x = 0 \) or \( x = -10 \).

For \( x = 0 \), \( y = 5 \) (point \( P_1(0, 5) \)).
For \( x = -10 \), \( y = -5 \) (point \( P_2(-10, -5) \)).

### Step 7: Calculate distances from the origin
Now we calculate the distances \( OP_1 \) and \( OP_2 \):

\[ OP_1 = \sqrt{0^2 + 5^2} = 5 \]
\[ OP_2 = \sqrt{(-10)^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \]

### Step 8: Find maximum and minimum distances
The maximum distance \( OP_{max} = 5\sqrt{5} \) and the minimum distance \( OP_{min} = 5 \).

### Step 9: Calculate the final value
Now we find:

\[ \frac{OP_{max} + OP_{min}}{2} = \frac{5\sqrt{5} + 5}{2} = \frac{5(\sqrt{5} + 1)}{2} \]

### Final Result:
The final answer is:

\[ \frac{5(\sqrt{5} + 1)}{2} \]