The equation `| z - omega |^(2) pm | z - omega ^(2)|^(2) = lambda ` represents the equation of a circle with ` omega , omega ^(2)` as the extremities of a diameter, then ` lambda ` is

Options

A. `sqrt(2)`
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the given equation and find the value of \( \lambda \) such that the equation represents a circle with \( \omega \) and \( \omega^2 \) as the extremities of a diameter.

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
The equation provided is:
\[ |z - \omega|^2 \pm |z - \omega^2|^2 = \lambda \]
Here, \( \omega \) and \( \omega^2 \) are the complex cube roots of unity, which can be expressed as:
\[ \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \quad \text{and} \quad \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \]

2. **Finding the Midpoint**:
The midpoint of the segment joining \( \omega \) and \( \omega^2 \) is given by:
\[ M = \frac{\omega + \omega^2}{2} \]
Calculating this:
\[ M = \frac{\left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) + \left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)}{2} = \frac{-1}{2} \]

3. **Finding the Radius**:
The distance between \( \omega \) and \( \omega^2 \) is the diameter of the circle. The radius \( r \) is half of this distance:
\[ d = |\omega - \omega^2| = |(-\frac{1}{2} + \frac{\sqrt{3}}{2}i) - (-\frac{1}{2} - \frac{\sqrt{3}}{2}i)| = |\sqrt{3}i| = \sqrt{3} \]
Therefore, the radius \( r \) is:
\[ r = \frac{d}{2} = \frac{\sqrt{3}}{2} \]

4. **Equation of the Circle**:
The standard form of a circle with center \( M \) and radius \( r \) is:
\[ |z - M|^2 = r^2 \]
Substituting \( M = -\frac{1}{2} \) and \( r = \frac{\sqrt{3}}{2} \):
\[ |z + \frac{1}{2}|^2 = \left(\frac{\sqrt{3}}{2}\right)^2 \]
This simplifies to:
\[ |z + \frac{1}{2}|^2 = \frac{3}{4} \]

5. **Relating to Given Equation**:
The equation \( |z - \omega|^2 + |z - \omega^2|^2 = \lambda \) can be manipulated to find \( \lambda \). Since we know the radius and the center, we can equate:
\[ \lambda = \frac{3}{4} \]

### Conclusion:
Thus, the value of \( \lambda \) is:
\[ \lambda = \frac{3}{4} \]