Let $c>0$. Then $$\min_{y}(ax^{2}+bxy+cy^{2})=\min_{y}\left( ax^{2}+\left( \sqrt{ c }y+\frac{b}{2\sqrt{ c }}x \right)^{2}-\frac{b^{2}}{4c}x^{2} \right)$$and the resultant minimizing $y$ is $$y^{*}=-\frac{b}{2c}x$$