If the equation `|z(z+1)^8 =z^8 |z+1|` where `z in C and z(z+1) != 0` has distinct roots `z_1,z_2,z_3,...,z_n`.(where `n in N`) then which of the following is/are true?

Options

A. `z_(1),z_(2),z_(3)……..z_(n)` are concyclic points
B. `z_(1),z_(2),z_(3),……...z_(n)` are collinear points
C. `sum_(r=1)^(n)"Re"(z_(r))=(-7)/2`
D. `sum_(r=1)^(n("Im")(z_(r))=0`

Explanation

To solve the equation \( |z(z+1)^8| = |z^8||z+1| \) where \( z \in \mathbb{C} \) and \( z(z+1) \neq 0 \), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ |z(z+1)^8| = |z^8||z+1| \]
This can be rewritten as:
\[ |z| \cdot |(z+1)^8| = |z^8| \cdot |z+1| \]

### Step 2: Simplify the Equation
Using the property of absolute values, we can express the equation as:
\[ |z| \cdot |z+1|^8 = |z|^8 \cdot |z+1| \]
Now, we can divide both sides by \( |z| \cdot |z+1| \) (since \( z(z+1) \neq 0 \)):
\[ |z+1|^7 = |z|^7 \]

### Step 3: Express in Terms of Modulus
This implies:
\[ \frac{|z+1|}{|z|} = 1 \]
Thus, we can write:
\[ |z+1| = |z| \]

### Step 4: Analyze the Implications
The equation \( |z+1| = |z| \) indicates that the points \( z \) and \( z+1 \) are equidistant from the origin. This means that the point \( z \) lies on the perpendicular bisector of the segment joining the points \( 0 \) and \( -1 \) in the complex plane.

### Step 5: Geometric Interpretation
The perpendicular bisector of the segment from \( 0 \) to \( -1 \) is the vertical line \( \text{Re}(z) = -\frac{1}{2} \). Thus, all solutions \( z \) must lie on this line.

### Step 6: Finding Distinct Roots
Since \( z \) is constrained to lie on the line \( \text{Re}(z) = -\frac{1}{2} \), we can express \( z \) as:
\[ z = -\frac{1}{2} + yi \quad (y \in \mathbb{R}) \]
Substituting this into our original equation will yield distinct roots based on the values of \( y \).

### Step 7: Conclusion
The distinct roots \( z_1, z_2, \ldots, z_n \) correspond to different values of \( y \) that satisfy the equation. The nature of these roots will depend on the specific conditions of the original equation.

### Final Answer
From the analysis, we conclude that the correct option is:
- The roots \( z_1, z_2, z_3, \ldots, z_n \) are collinear points on the line \( \text{Re}(z) = -\frac{1}{2} \).