Question

Let $\alpha ,\beta ,\gamma$ are the real roots of the equation $x^3+a{x}^2+bx+c=0(a,b,c\in Randa\ne 0)$. If the system of equations (in $u,v,w$) given by $\alpha u+\beta v+\gamma w=0,\beta u+\gamma v+\alpha w=0,\alpha \gamma u+\alpha v+\beta w=0$ has non-trivial solutions, then the value of $a^2/b$ is ...................

Answer 1

Given $\alpha ,\beta ,\gamma$ are the real roots of the equation $x^3+a{x}^2+bx+c=0(a,b,c\in Randa\ne 0)$.
And the system of equations (in $u,v,w$) given by $\alpha u+\beta v+\gamma w=0,\beta u+\gamma v+\alpha w=0,\alpha \gamma u+\alpha v+\beta w=0$ has non-trivial solution.
$\Rightarrow \left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=0$
$\Rightarrow (\alpha +\beta +\gamma )(\alpha \beta +\beta \gamma +\gamma \alpha -{\alpha }^2-{\beta }^2-{\gamma }^2)=0$
$\Rightarrow (\alpha +\beta +\gamma )\left(3\right(\alpha \beta +\beta \gamma +\gamma \alpha )-(\alpha +\beta +\gamma {)}^2)=0$
$\Rightarrow -a(3b-a^2)=0$
$\therefore \frac{a^2}{b}=3$ ($\because a\ne 0$)