To solve the problem, we will analyze both statements step by step.
### Step 1: Analyze Statement 1
We are given that \( a, b, c \) are three non-zero real numbers such that:
\[
a + b + c = 0
\]
This implies:
\[
c = - (a + b)
\]
We also have three complex numbers \( z_1, z_2, z_3 \) such that:
\[
a z_1 + b z_2 + c z_3 = 0
\]
Substituting \( c \) into the equation:
\[
a z_1 + b z_2 - (a + b) z_3 = 0
\]
Rearranging gives:
\[
a z_1 + b z_2 = (a + b) z_3
\]
### Step 2: Express \( z_3 \)
From the rearranged equation, we can express \( z_3 \) as:
\[
z_3 = \frac{a z_1 + b z_2}{a + b}
\]
This shows that \( z_3 \) is a linear combination of \( z_1 \) and \( z_2 \).
### Step 3: Ratio Interpretation
The equation indicates that \( z_3 \) divides the line segment joining \( z_1 \) and \( z_2 \) in the ratio \( a:b \). This means that \( z_1, z_2, z_3 \) are collinear points.
### Step 4: Circle Condition
For three points to lie on a circle, they must not be collinear. Since we have established that \( z_1, z_2, z_3 \) are collinear, Statement 1 is false.
### Conclusion for Statement 1
Thus, Statement 1 is **false**.
---
### Step 5: Analyze Statement 2
Statement 2 states that if \( z_1, z_2, z_3 \) are collinear, then:
\[
\left| \begin{array}{ccc}
z_1 & \overline{z_1} & 1 \\
z_2 & \overline{z_2} & 1 \\
z_3 & \overline{z_3} & 1
\end{array} \right| = 0
\]
### Step 6: Determinant Calculation
If \( z_1, z_2, z_3 \) are collinear, the determinant will be zero. This is because the rows of the determinant represent points in the complex plane, and if they are collinear, the area formed by these points is zero.
### Step 7: Example with Collinear Points
Assuming \( z_1 = x_1 + i0 \), \( z_2 = x_2 + i0 \), \( z_3 = x_3 + i0 \) (all on the x-axis), we can write:
\[
\left| \begin{array}{ccc}
x_1 & x_1 & 1 \\
x_2 & x_2 & 1 \\
x_3 & x_3 & 1
\end{array} \right| = 0
\]
This is because two columns are identical, leading to a determinant of zero.
### Conclusion for Statement 2
Thus, Statement 2 is **true**.
---
### Summary
- Statement 1 is false because \( z_1, z_2, z_3 \) are collinear, hence cannot lie on a circle.
- Statement 2 is true as the determinant condition holds for collinear points.
---
