Locating a shortest vector in certain $2$-dimensional lattices

Let $a$, $m$ be positive integers, $1<a<m$, $\gcd (a,m)=1$. We determine the location of a shortest vector in the $2$-dimensional lattices $$\mathrm{\Lambda }(a,m)=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}\mid ax+y\equiv 0(m)\}\mathrm{.}$$ This confirms a conjecture of Han Wu and Guangwu Xu.