If `z^4= (z-1)^4` then the roots are represented in the Argand plane by the points that are

Options

A. collinear
B. concyclic
C. vertices of a parallelogram
D. none of these

Answer

To solve the equation \( z^4 = (z - 1)^4 \), we will follow these steps:

### Step 1: Take the fourth root of both sides
We start with the equation:
\[ z^4 = (z - 1)^4 \]
Taking the fourth root of both sides gives us:
\[ z - 1 = \pm z \]

### Step 2: Solve for \( z \)
This leads us to two cases:

**Case 1:**
\[ z - 1 = z \]
This simplifies to:
\[ -1 = 0 \]
which has no solution.

**Case 2:**
\[ z - 1 = -z \]
Rearranging gives:
\[ 2z = 1 \implies z = \frac{1}{2} \]

### Step 3: Solve for the imaginary cases
Now we consider the imaginary roots. We can express \( z \) in terms of \( i \) (the imaginary unit):
\[ z - 1 = i z \quad \text{and} \quad z - 1 = -i z \]

**For \( z - 1 = i z \):**
\[ z - i z = 1 \implies z(1 - i) = 1 \implies z = \frac{1}{1 - i} \]
To simplify \( \frac{1}{1 - i} \), we multiply the numerator and denominator by the conjugate:
\[ z = \frac{1(1 + i)}{(1 - i)(1 + i)} = \frac{1 + i}{1 + 1} = \frac{1 + i}{2} = \frac{1}{2} + \frac{1}{2} i \]

**For \( z - 1 = -i z \):**
\[ z + i z = 1 \implies z(1 + i) = 1 \implies z = \frac{1}{1 + i} \]
Similarly, we simplify:
\[ z = \frac{1(1 - i)}{(1 + i)(1 - i)} = \frac{1 - i}{1 + 1} = \frac{1 - i}{2} = \frac{1}{2} - \frac{1}{2} i \]

### Step 4: Collect all solutions
We have found three solutions:
1. \( z = \frac{1}{2} \)
2. \( z = \frac{1}{2} + \frac{1}{2} i \)
3. \( z = \frac{1}{2} - \frac{1}{2} i \)

### Step 5: Analyze the points in the Argand plane
Now we plot these points in the Argand plane:
- \( z = \frac{1}{2} \) corresponds to the point \( \left( \frac{1}{2}, 0 \right) \)
- \( z = \frac{1}{2} + \frac{1}{2} i \) corresponds to the point \( \left( \frac{1}{2}, \frac{1}{2} \right) \)
- \( z = \frac{1}{2} - \frac{1}{2} i \) corresponds to the point \( \left( \frac{1}{2}, -\frac{1}{2} \right) \)

All three points lie on the vertical line \( x = \frac{1}{2} \), indicating that they are collinear.

### Conclusion
Thus, the roots represented in the Argand plane by the points are collinear.

---