$\therefore D = 0 = \left| {\begin{array}{*{20}{c}}
2&3&{ - 1}\\
1&k&{ - 2}\\
2&{ - 1}&1
\end{array}} \right| = 0$

$ \Rightarrow k = \frac{9}{2}$

So equation are $2x + 3y - z = 0\,\,\,\,\,\,\,\,\,.....\left( 1 \right)$

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

                          $2x - y + z = 0\,\,\,\,\,\,\,\,\,\,\,\,....\left( 3 \right)$

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

$ \Rightarrow 2y = z\,\,\,\,\,\,\,\,\,....\left( 4 \right)$

$ \Rightarrow \frac{y}{z} = \frac{1}{2}$

From equation $(1)$ and $(4)$

$2x + 3y - 2y = 0$

$ \Rightarrow 2y + y = 0$

$ \Rightarrow \frac{x}{y} = \frac{{ - 1}}{2}$ or $\frac{z}{x} =  - 4$

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