The locus of any point `P(z)` on argand plane is `arg((z-5i)/(z+5i))=(pi)/(4)`.
Total number of integral points inside the region bounded by the locus of `P(z)` and imaginery axis on the argand plane is

Options

A. `62`
B. `74`
C. `136`
D. `138`

Explanation

To solve the problem, we need to analyze the given expression and find the locus of the point \( P(z) \) on the Argand plane.

### Step-by-Step Solution:

1. **Understanding the Argument Condition**:
We start with the equation given in the problem:
\[ \arg\left(\frac{z - 5i}{z + 5i}\right) = \frac{\pi}{4} \]
This means that the angle formed by the line connecting the point \( z \) to \( 5i \) and the line connecting \( z \) to \( -5i \) is \( \frac{\pi}{4} \).

2. **Setting \( z \)**:
Let \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number.

3. **Substituting \( z \)**:
Substitute \( z \) into the argument condition:
\[ \arg\left(\frac{(x + yi) - 5i}{(x + yi) + 5i}\right) = \arg\left(\frac{x + (y - 5)i}{x + (y + 5)i}\right) \]

4. **Finding the Argument**:
The argument of a complex number \( a + bi \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \). Therefore, we can express the arguments:
\[ \arg\left(x + (y - 5)i\right) = \tan^{-1}\left(\frac{y - 5}{x}\right) \]
\[ \arg\left(x + (y + 5)i\right) = \tan^{-1}\left(\frac{y + 5}{x}\right) \]

5. **Setting Up the Equation**:
The equation becomes:
\[ \tan^{-1}\left(\frac{y - 5}{x}\right) - \tan^{-1}\left(\frac{y + 5}{x}\right) = \frac{\pi}{4} \]

6. **Using the Tangent Difference Formula**:
Using the tangent difference formula:
\[ \tan\left(\tan^{-1}(A) - \tan^{-1}(B)\right) = \frac{A - B}{1 + AB} \]
where \( A = \frac{y - 5}{x} \) and \( B = \frac{y + 5}{x} \).

7. **Solving the Equation**:
This leads to:
\[ \frac{\frac{y - 5}{x} - \frac{y + 5}{x}}{1 + \frac{(y - 5)(y + 5)}{x^2}} = 1 \]
Simplifying gives:
\[ \frac{-10/x}{1 + \frac{y^2 - 25}{x^2}} = 1 \]
Cross-multiplying and simplifying leads to:
\[ -10 = x^2 + y^2 - 25 \]
Rearranging gives:
\[ x^2 + y^2 = 15 \]

8. **Identifying the Circle**:
This represents a circle centered at the origin with a radius of \( \sqrt{15} \).

9. **Finding Integral Points**:
We need to find the integer points inside this circle. The maximum integer values for \( x \) are \( -3, -2, -1, 0, 1, 2, 3 \).

10. **Counting Integral Points**:
For each integer \( x \) from \( -3 \) to \( 3 \), calculate the corresponding \( y \) values:
- For \( x = 0 \): \( y^2 < 15 \) gives \( y = -3, -2, -1, 0, 1, 2, 3 \) (7 points).
- For \( x = 1 \) and \( x = -1 \): \( y^2 < 14 \) gives \( y = -3, -2, -1, 0, 1, 2, 3 \) (7 points each).
- For \( x = 2 \) and \( x = -2 \): \( y^2 < 11 \) gives \( y = -3, -2, -1, 0, 1, 2, 3 \) (7 points each).
- For \( x = 3 \) and \( x = -3 \): \( y^2 < 6 \) gives \( y = -2, -1, 0, 1, 2 \) (5 points each).

11. **Total Count**:
Adding all these points:
\[ 7 + 7 + 7 + 7 + 5 + 5 = 38 \]

### Final Answer:
The total number of integral points inside the region bounded by the locus of \( P(z) \) and the imaginary axis is **38**.