  For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in
the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT
stability of binary forms of degree $2a$. Moreover, we prove that such a
hypersurface is K-polystable and not K-stable if it is quasi-smooth.
