If the system of equations $2x+3y-z=0,x+ky-2z=0$and $2x-y+z=0$has a non trivial solution. Then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k=$

$\left|\begin{array}{ccc}2& 3& -1\\ 1& k& -2\\ 2& -1& 1\end{array}\right|=0\Rightarrow K=\frac{9}{2}$

Given equations are $2x+3y-z=0\to \left(1\right)$

$x+\frac{9}{2}y-2z=0\to \left(2\right),2x-y+z=0\to \left(3\right)$

$\left(1\right)-\left(3\right)\Rightarrow 4y-2z=0\Rightarrow \frac{z}{y}=2$

$\left(1\right)+\left(3\right)\Rightarrow \frac{x}{y}=\frac{-1}{2}\to \left(5\right)$

$\left(4\right)\times \left(5\right)\Rightarrow \frac{y}{z}\times \frac{x}{y}=\frac{-1}{4}\Rightarrow \frac{z}{x}=-4$

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k=-\frac{1}{2}+\frac{1}{2}-4+\frac{9}{2}=\frac{1}{2}$