If `omega ne 1` is a cube root of unity and `|z-1|^(2) + 2|z-omega|^(2) = 3|z - omega^(2)|^(2)` then z lies on

Options

A. a straight line
B. a parabola
C. an ellipse
D. a rectangular hyperbola

Answer

To solve the equation \( |z - 1|^2 + 2|z - \omega|^2 = 3|z - \omega^2|^2 \), where \(\omega\) is a cube root of unity and \(\omega \neq 1\), we can follow these steps:

### Step 1: Express the Modulus Squared
We start by expanding each term in the equation using the definition of modulus squared:
1. \( |z - 1|^2 = (z - 1)(\overline{z} - 1) = z\overline{z} - z - \overline{z} + 1 \)
2. \( |z - \omega|^2 = (z - \omega)(\overline{z} - \overline{\omega}) = z\overline{z} - z\overline{\omega} - \overline{z}\omega + \omega\overline{\omega} \)
3. \( |z - \omega^2|^2 = (z - \omega^2)(\overline{z} - \overline{\omega^2}) = z\overline{z} - z\overline{\omega^2} - \overline{z}\omega^2 + \omega^2\overline{\omega^2} \)

### Step 2: Substitute and Simplify
Substituting these expressions into the original equation:
\[ (z\overline{z} - z - \overline{z} + 1) + 2(z\overline{z} - z\overline{\omega} - \overline{z}\omega + 1) = 3(z\overline{z} - z\overline{\omega^2} - \overline{z}\omega^2 + 1) \]
This simplifies to:
\[ z\overline{z} - z - \overline{z} + 1 + 2z\overline{z} - 2z\overline{\omega} - 2\overline{z}\omega + 2 = 3z\overline{z} - 3z\overline{\omega^2} - 3\overline{z}\omega^2 + 3 \]

### Step 3: Rearranging the Equation
Combine like terms:
\[ 3z\overline{z} - z - \overline{z} + 3 - 2z\overline{\omega} - 2\overline{z}\omega = 3z\overline{z} - 3z\overline{\omega^2} - 3\overline{z}\omega^2 + 3 \]
This leads to:
\[ -z - \overline{z} + 3 - 2z\overline{\omega} - 2\overline{z}\omega = -3z\overline{\omega^2} - 3\overline{z}\omega^2 \]

### Step 4: Isolate Terms
Rearranging gives:
\[ z + \overline{z} + 2z\overline{\omega} + 2\overline{z}\omega = 3z\overline{\omega^2} + 3\overline{z}\omega^2 \]

### Step 5: Identify the Locus
This equation describes a relationship between \(z\) and its conjugate, indicating that \(z\) lies on a specific geometric locus. To determine the exact nature of this locus, we can analyze the coefficients and the terms involved.

### Conclusion
The final form of the equation suggests that \(z\) lies on a **rectangular hyperbola**.