Let z be a non-zero complex number such that area of the triangle with vertices `P(z), Q(ze^(pii//6)) and R(z(1+e^(pii//6)))` is 0.64, then z lies on a circle of diameter ___________.

Answer

To solve the problem, we need to determine the diameter of the circle on which the complex number \( z \) lies, given that the area of the triangle formed by the points \( P(z) \), \( Q(ze^{\frac{\pi i}{6}}) \), and \( R(z(1+e^{\frac{\pi i}{6}})) \) is 0.64.

### Step-by-Step Solution:

1. **Understanding the Area of the Triangle**:
The area \( A \) of a triangle formed by three points in the complex plane can be calculated using the formula:
\[ A = \frac{1}{2} \left| z_1(z_2^* - z_3^*) + z_2(z_3^* - z_1^*) + z_3(z_1^* - z_2^*) \right| \]
where \( z_1, z_2, z_3 \) are the vertices of the triangle.

2. **Identifying the Points**:
Here, we have:
- \( z_1 = z \)
- \( z_2 = ze^{\frac{\pi i}{6}} \)
- \( z_3 = z(1 + e^{\frac{\pi i}{6}}) \)

3. **Calculating the Area**:
We know the area is given as 0.64. Thus, we set up the equation:
\[ \frac{1}{2} \left| z(z e^{-\frac{\pi i}{6}} - z(1 + e^{-\frac{\pi i}{6}})) + ze^{\frac{\pi i}{6}}(z(1 + e^{-\frac{\pi i}{6}}) - z) + z(1 + e^{\frac{\pi i}{6}})(z - ze^{\frac{\pi i}{6}}) \right| = 0.64 \]

4. **Simplifying the Expression**:
This expression simplifies to:
\[ \frac{1}{2} \left| z \left( z e^{-\frac{\pi i}{6}} - z - z e^{-\frac{\pi i}{6}} + z \right) + ze^{\frac{\pi i}{6}}(z(1 + e^{-\frac{\pi i}{6}}) - z) + z(1 + e^{\frac{\pi i}{6}})(z - ze^{\frac{\pi i}{6}}) \right| \]
After simplification, we can find that:
\[ \text{Area} = \frac{1}{4} |z|^2 \left| e^{\frac{\pi i}{6}} - 1 \right| \]

5. **Finding the Magnitude**:
The magnitude \( |e^{\frac{\pi i}{6}} - 1| \) can be calculated as:
\[ |e^{\frac{\pi i}{6}} - 1| = \sqrt{(cos(\frac{\pi}{6}) - 1)^2 + (sin(\frac{\pi}{6}))^2} = \sqrt{(\frac{\sqrt{3}}{2} - 1)^2 + (\frac{1}{2})^2} \]

6. **Setting Up the Equation**:
From the area, we have:
\[ \frac{1}{4} |z|^2 \cdot |e^{\frac{\pi i}{6}} - 1| = 0.64 \]
Therefore,
\[ |z|^2 = \frac{4 \cdot 0.64}{|e^{\frac{\pi i}{6}} - 1|} \]

7. **Finding the Diameter**:
The diameter \( D \) of the circle on which \( z \) lies is given by:
\[ D = 2|z| = 2\sqrt{|z|^2} \]

8. **Final Calculation**:
Substitute the value of \( |z|^2 \) into the diameter formula to find the final answer.