If n is a natural number `gt 2`, such that `z^(n) = (z+1)^(n)`, then (a) roots of equation lie on a straight line parallel to the y-axis (b) roots of equaiton lie on a straight line parallel to the x-axis (c) sum of the real parts of the roots is `-[(n-1)//2]` (d) none of these

Options

A. roots of equation lie on a straight line parallel to the y-axis
B. roots of equaiton lie on a straight line parallel to the x-axis
C. sum of the real parts of the roots is `-[(n-1)//2]`
D. none of these

Explanation

To solve the equation \( z^n = (z + 1)^n \) where \( n \) is a natural number greater than 2, we can follow these steps:

### Step 1: Take the modulus of both sides
We start with the equation:
\[ z^n = (z + 1)^n \]
Taking the modulus on both sides gives us:
\[ |z|^n = |z + 1|^n \]

### Step 2: Simplify the equation
Since \( n \) is a natural number, we can take the \( n \)-th root of both sides:
\[ |z| = |z + 1| \]

### Step 3: Square both sides
Next, we square both sides to eliminate the modulus:
\[ |z|^2 = |z + 1|^2 \]

### Step 4: Substitute \( z \) as a complex number
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. Then we have:
\[ |z|^2 = x^2 + y^2 \]
and
\[ |z + 1|^2 = |(x + 1) + iy|^2 = (x + 1)^2 + y^2 \]
So, we can write:
\[ x^2 + y^2 = (x + 1)^2 + y^2 \]

### Step 5: Cancel out \( y^2 \)
Subtract \( y^2 \) from both sides:
\[ x^2 = (x + 1)^2 \]

### Step 6: Expand and simplify
Expanding the right side gives:
\[ x^2 = x^2 + 2x + 1 \]
Subtract \( x^2 \) from both sides:
\[ 0 = 2x + 1 \]

### Step 7: Solve for \( x \)
Solving for \( x \) gives:
\[ 2x = -1 \implies x = -\frac{1}{2} \]

### Step 8: Analyze the roots
Since \( x = -\frac{1}{2} \), the real part of \( z \) is constant. The imaginary part \( y \) can take any value, which means the roots lie on the line \( x = -\frac{1}{2} \).

### Step 9: Sum of the real parts of the roots
Since there are \( n - 1 \) roots (as \( z^n = (z + 1)^n \) implies \( n \) roots), the sum of the real parts of the roots will be:
\[ \text{Sum of real parts} = (n - 1) \cdot \left(-\frac{1}{2}\right) = -\frac{n - 1}{2} \]

### Conclusion
Thus, we conclude:
- The roots of the equation lie on a straight line parallel to the y-axis (since \( x = -\frac{1}{2} \)).
- The sum of the real parts of the roots is \(-\frac{n - 1}{2}\).

### Final Answer
The correct options are:
(a) Roots of the equation lie on a straight line parallel to the y-axis.
(c) Sum of the real parts of the roots is \(-\frac{n - 1}{2}\).