The mirror image of the curve `arg((z-3)/(z-i))=pi/6, i=sqrt(- 1)` in the real axis

Options

A. `arg((z+3)/(z+i))=(pi)/(6)`
B. `arg((z-3)/(z+i))=(pi)/(6)`
C. `arg((z+i)/(z+3))=(pi)/(6)`
D. `arg((z+i)/(z-3))=(pi)/(6)`

Answer

To find the mirror image of the curve defined by the equation \( \arg\left(\frac{z-3}{z-i}\right) = \frac{\pi}{6} \) in the real axis, we can follow these steps:

### Step 1: Understand the Given Curve
The given equation represents a curve in the complex plane. The expression \( z \) can be represented as \( z = x + iy \), where \( x \) and \( y \) are the real and imaginary parts, respectively.

### Step 2: Set Up the Mirror Image
The mirror image of a point \( z \) in the real axis is given by its complex conjugate \( \overline{z} \). Therefore, we can express the mirror image of the curve as:
\[ \arg\left(\frac{\overline{z} - 3}{\overline{z} - i}\right) = \frac{\pi}{6} \]

### Step 3: Substitute the Conjugate
Substituting \( \overline{z} = x - iy \) into the equation, we have:
\[ \arg\left(\frac{(x - iy) - 3}{(x - iy) - i}\right) = \frac{\pi}{6} \]
This simplifies to:
\[ \arg\left(\frac{(x - 3) - iy}{(x) - (y + 1)i}\right) = \frac{\pi}{6} \]

### Step 4: Express the Argument
The argument of a complex number \( \frac{a + bi}{c + di} \) can be expressed as:
\[ \arg\left(\frac{a + bi}{c + di}\right) = \tan^{-1}\left(\frac{b}{a}\right) - \tan^{-1}\left(\frac{d}{c}\right) \]
In our case, we can express the argument as:
\[ \arg\left((x - 3) - iy\right) - \arg\left(x - (y + 1)i\right) = \frac{\pi}{6} \]

### Step 5: Solve for the Argument
Calculating the arguments, we have:
\[ \tan^{-1}\left(\frac{-y}{x - 3}\right) - \tan^{-1}\left(\frac{-(y + 1)}{x}\right) = \frac{\pi}{6} \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ \tan^{-1}\left(\frac{-y}{x - 3}\right) = \tan^{-1}\left(\frac{-(y + 1)}{x}\right) + \frac{\pi}{6} \]

### Step 7: Final Form
The final form of the equation represents the mirror image of the original curve in the real axis.

### Summary
The mirror image of the curve \( \arg\left(\frac{z-3}{z-i}\right) = \frac{\pi}{6} \) in the real axis is given by:
\[ \arg\left(\frac{\overline{z} - 3}{\overline{z} - i}\right) = \frac{\pi}{6} \]