Descartes folium

Question

The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve: $$bx^3+y^3=3axy$$ The previous curve is a cubic curve with a node at the origin and their tangent directions are de coordenate axis... (the property I'm looking for is in terms of the coefficients a,b)

Answer 1

You can change your coordinates and the coefficient: $$bx^3+y^3=3axy$$ $$(\sqrt[3]{b}x)^3+y^3=3a\frac{1}{\sqrt[3]{b}}(\sqrt[3]{b}x)y$$ $$\sqrt[3]{b}x\mapsto x_1\qquad y\mapsto y_1\qquad \frac{a}{\sqrt[3]{b}}\mapsto a_1$$ So you have a the usual form: $$x_1^3+y_1^3=3a_1x_1y_1$$

So if in general you know a property of type: $$f(x,y,a)=0$$ for the first curve, you can transform it in an equivalent property of type: $$f\left(\sqrt[3]{b}x_1,y_1,\frac{a}{\sqrt[3]{b}}\right)=0$$ for the second curve.