Problem 549

Given vectors $\vec{a}=\{1, \sqrt{2}\}, \vec{b}=\{-\sqrt{2}, 1\}$. If positive numbers $k$ and $t$ make
$$\vec{x}=\vec{a}+\left(t^{2}+1\right) \vec{b} \text { and } \vec{y}=-k \vec{a}+\frac{1}{t} \vec{b}$$

perpendicular, then the minimum value of $k$ is