If arg `((z-(10+6i))/(z-(4+2i)))=(pi)/(4)` (where z is a complex number), then the perimeter of the locus of z is

Options

A. `(sqrt(13)pi)/(4)" units"`
B. `(3sqrt(13)pi)/(4)" units"`
C. `3sqrt(13)pi" units"`
D. `(3pi)/(2)sqrt(26)" units"`

Answer

To solve the problem, we need to analyze the given equation involving complex numbers and their arguments. Let's break it down step by step.

### Step 1: Understand the Given Equation
We are given:
\[ \arg\left(\frac{z - (10 + 6i)}{z - (4 + 2i)}\right) = \frac{\pi}{4} \]
This means that the angle formed by the line segments from \( z \) to the points \( z_1 = 10 + 6i \) and \( z_2 = 4 + 2i \) is \( \frac{\pi}{4} \) radians (or 45 degrees).

### Step 2: Geometric Interpretation
The argument condition implies that the point \( z \) lies on a circular arc where the angle between the lines connecting \( z \) to \( z_1 \) and \( z_2 \) is \( \frac{\pi}{4} \).

### Step 3: Determine the Points
Identify the points:
- \( z_1 = 10 + 6i \) corresponds to the point \( (10, 6) \).
- \( z_2 = 4 + 2i \) corresponds to the point \( (4, 2) \).

### Step 4: Calculate the Distance Between \( z_1 \) and \( z_2 \)
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(4 - 10)^2 + (2 - 6)^2} = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

### Step 5: Find the Radius of the Circle
The angle \( \frac{\pi}{4} \) implies that the locus of points forms a circular arc. The angle subtended at the center of the circle by the chord \( z_1 z_2 \) is \( \frac{\pi}{2} \) (since \( 2 \times \frac{\pi}{4} = \frac{\pi}{2} \)).

Using the property of circles, the radius \( r \) can be found using the relationship:
\[ r = \frac{d}{2 \sin(\theta/2)} \]
where \( \theta = \frac{\pi}{2} \) and \( d = 2\sqrt{13} \).

Thus,
\[ r = \frac{2\sqrt{13}}{2 \sin(\frac{\pi}{4})} = \frac{2\sqrt{13}}{2 \cdot \frac{\sqrt{2}}{2}} = \frac{\sqrt{13}}{\frac{\sqrt{2}}{2}} = \sqrt{13} \cdot \frac{2}{\sqrt{2}} = \sqrt{26} \]

### Step 6: Calculate the Perimeter of the Locus
The locus is a quarter circle (since the angle is \( \frac{\pi}{2} \)). The full circumference of a circle is given by \( 2\pi r \). Therefore, for a quarter circle:
\[ \text{Perimeter} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]
Substituting \( r = \sqrt{26} \):
\[ \text{Perimeter} = \frac{\pi \sqrt{26}}{2} \]

### Final Answer
The perimeter of the locus of \( z \) is:
\[ \frac{3\pi \sqrt{26}}{2} \text{ units} \]