2019 RI P1 Q5

The diagram shows the graph of Folium of Descartes with cartesian equation

\({{x}^{3}}+{{y}^{3}}=3axy\),

where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation \(y=-x-a\).

1. Given that \(\left( 0,0 \right)\) is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.[5]
2. Sketch the graph of
   \({{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y\),
   including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the \(x\)-axis.[3]