Let system of equation `alphau+betav+gammaw=0 , betau+gammav+alphaw=0 , gammau+alphav+betaw=0` has non trial solution and `alpha, beta, gamma` are distinct root `x^3+ax^2+bx+c=0` then find value of `a^2/b`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the given system of equations and the properties of the roots of the polynomial. Let's break it down step by step.

### Step 1: Understanding the System of Equations
We have the following system of equations:
1. \( \alpha u + \beta v + \gamma w = 0 \)
2. \( \beta u + \gamma v + \alpha w = 0 \)
3. \( \gamma u + \alpha v + \beta w = 0 \)

For this system to have a non-trivial solution (i.e., not all variables are zero), the determinant of the coefficients must be zero.

### Step 2: Forming the Coefficient Matrix
The coefficient matrix \( A \) of the system can be written as:
\[ A = \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \]

### Step 3: Calculating the Determinant
We need to calculate the determinant of matrix \( A \) and set it to zero:
\[ \text{det}(A) = \begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix} \]

Using the determinant formula for a \( 3 \times 3 \) matrix, we can expand it:
\[ \text{det}(A) = \alpha(\gamma^2 - \alpha\beta) - \beta(\beta\gamma - \alpha^2) + \gamma(\beta^2 - \gamma\alpha) \]

### Step 4: Setting the Determinant to Zero
Setting the determinant to zero gives us:
\[ \alpha(\gamma^2 - \alpha\beta) - \beta(\beta\gamma - \alpha^2) + \gamma(\beta^2 - \gamma\alpha) = 0 \]

### Step 5: Using the Roots of the Polynomial
Given that \( \alpha, \beta, \gamma \) are distinct roots of the polynomial \( x^3 + ax^2 + bx + c = 0 \), we can use Vieta's formulas:
- \( \alpha + \beta + \gamma = -a \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = b \)
- \( \alpha\beta\gamma = -c \)

### Step 6: Finding \( a \)
From the condition that the sum of the roots \( \alpha + \beta + \gamma = 0 \), we have:
\[ -a = 0 \implies a = 0 \]

### Step 7: Finding \( a^2 / b \)
Now we need to find \( \frac{a^2}{b} \):
\[ \frac{a^2}{b} = \frac{0^2}{b} = 0 \]

### Conclusion
Thus, the value of \( \frac{a^2}{b} \) is:
\[ \boxed{0} \]