If $abc$ and the system of equations $a x+b y+c z=0, b x+c y+a z=0$ and $c x+a y+b z=0$ has a non

Question

If $a>b>c$ and the system of equations $a x+b y+c z=0, b x+c y+a z=0$ and $c x+a y+b z=0$ has a non trivial solution then both the roots of equation $a t^2+b t+c=0$ are

Professor PC Thomas Classes · Answer

$ \begin{aligned} & \Delta=\left|\begin{array}{lll} a & b & c \ b & c & a \ c & a & b \end{array} ight|=0 \ & \Rightarrow a+b+c=0 ext { or } a=b=c \ & a>b>c \Rightarrow a+b+c=0 \Rightarrow a>0 \& C<0 \ & \Rightarrow 1 ext { is a root of } a t^2+b t+c=0 \ & ext { And other root is } \frac{c}{a}<0 \end{aligned} $