Let $\alpha ,\beta ,\gamma$ are the real roots of the equation $x^3+a{x}^2+bx+c=0(a,b,c\in R\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}a\ne 0)$. If the system of equations (in $u,v$ and $w$ ) given by

$$\begin{gathered} \alpha u+\beta v+\gamma w=0, \\ \beta u+\gamma v+\alpha w=0, \\ \gamma u+\alpha v+\beta w=0, \end{gathered}$$

has non trivial solutions, then the value of  $\frac{a^2}{b}$ is _____

Equation $x^3+a{x}^2+bx+c=0$ has roots $\alpha ,\beta ,\gamma .$ Therefore, 
 $$\begin{gathered} \alpha +\beta +\gamma =-a \\ \alpha \beta +\beta \gamma +\gamma \alpha =b \end{gathered}$$
Since the given system of equations has non-trivial solutions, we have
$$\begin{gathered} \left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=0 \\ or {\alpha }^3+{\beta }^3+{\gamma }^3-3\alpha \beta \gamma =0 \\ or (\alpha +\beta +\gamma )[{\alpha }^2+{\beta }^2+{\gamma }^2-\alpha \beta -\beta \gamma -\gamma \alpha ]=0 \\ or (\alpha +\beta +\gamma )[{(\alpha +\beta +\gamma )}^2-3(\alpha \beta +\beta \gamma +\gamma \alpha )]=0 \\ \Rightarrow -a[a^2-3b]=0 \\ or {\text{a}}^{\text{2}}\text{/\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}3} \end{gathered}$$