Let α, β, γ be the real roots of the equation, x^3+ a x^2+ b x+ c =0,( a, b, c ∈ R and a, b ≠ 0). If the system of equations (in u, v, w ) given by α…

Let $\displaystyle \alpha, \beta, \gamma$ be the real roots of the equation, $\displaystyle x^3+\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}=0,(\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{R}$ and $\displaystyle \mathrm{a}, \mathrm{b} \neq 0)$. If the system of equations (in $\displaystyle u, v, w$ ) given by $\displaystyle \alpha u+\beta v+\gamma w=0 ; \beta u+\gamma v+\alpha w=0$; $\displaystyle \gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\displaystyle \frac{\mathrm{a}^2}{\mathrm{~b}}$ is :