If `z_1,z_2,z_3` are any three roots of the equation `z^6=(z+1)^6,` then `arg((z_1-z_3)/(z_2-z_3))` can be equal to

Options

A. `0`
B. `pi`
C. `(pi)/(4)`
D. `-(pi)/(4)`

Explanation

To solve the problem, we start with the equation given:

### Step 1: Rewrite the equation
We have the equation:
\[ z^6 = (z + 1)^6 \]
Taking the sixth root of both sides, we can express this as:
\[ z = (z + 1) e^{i \frac{2k\pi}{6}} \quad \text{for } k = 0, 1, 2, 3, 4, 5 \]

### Step 2: Simplify the equation
Rearranging gives:
\[ z - (z + 1)e^{i \frac{2k\pi}{6}} = 0 \]
This can be simplified to:
\[ z(1 - e^{i \frac{2k\pi}{6}}) = e^{i \frac{2k\pi}{6}} \]

### Step 3: Find the modulus
Taking the modulus of both sides, we have:
\[ |z|^6 = |z + 1|^6 \]
This implies:
\[ |z| = |z + 1| \]
This means that the distance from \( z \) to the origin (0,0) is equal to the distance from \( z \) to the point (-1,0).

### Step 4: Determine the locus of points
The locus of points that are equidistant from the points (0,0) and (-1,0) is the perpendicular bisector of the segment joining these two points. The equation of this line is:
\[ x = -\frac{1}{2} \]
This means that all roots \( z_1, z_2, z_3 \) lie on this vertical line.

### Step 5: Analyze the argument
Since \( z_1, z_2, z_3 \) are collinear on the vertical line \( x = -\frac{1}{2} \), we can express the argument:
\[ \arg\left(\frac{z_1 - z_3}{z_2 - z_3}\right) \]
Since all points are on the same vertical line, the difference \( z_1 - z_3 \) and \( z_2 - z_3 \) will also be vertical. Therefore, the argument will be either \( 0 \) or \( \pi \) depending on the order of \( z_1, z_2, z_3 \).

### Conclusion
Thus, the possible values for \( \arg\left(\frac{z_1 - z_3}{z_2 - z_3}\right) \) can be:
\[ 0, \pi, 2\pi, \ldots \quad (2k\pi \text{ for } k \in \mathbb{Z}) \]

### Final Answer
The argument can be equal to \( 0 \) or \( \pi \).

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