A curve \(C\) has equation \(x^{2}+4 x y-y^{2}+20=0\). Show that, at stationary points on \(C, x=-2 y\).

Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the stationary points.