All the three equations obtained are
$\left(\cos \alpha \right)x+\left(\sin \alpha \right)z=0$
$y=1$
$\left(\sin \alpha \right)x+\left(\cos \alpha \right)z=0$
So, the equations $\left(\cos \alpha \right)x+\left(\sin \alpha \right)z=0$ and $\left(\sin \alpha \right)x+\left(\cos \alpha \right)z=0$ has infinite solutions
Then, $∆=\left|\begin{array}{c}\cos \alpha \\ \sin \alpha \end{array}\begin{array}{c}\sin \alpha \\ \cos \alpha \end{array}\right|=0$

$$\begin{gathered} \Rightarrow {\cos }^2\alpha -{\sin }^2\alpha =0 \\ \Rightarrow \cos 2\alpha =0 \end{gathered}$$
$\Rightarrow \alpha =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$
Sum of all values of $\alpha$ $=4\pi$

Hence, $k=4$