The folium of Descartes is the curve with equation x³ + y³ = 3axy, where a 0 is a constant (Figure 25).

(a) Show that the line y = tx intersects the folium at the origin and at one other point P for all t ≠ −1, 0. Express the coordinates of P in terms of t to obtain a parametrization of the folium. Indicate the direction of the parametrization on the graph.

(b) Describe the interval of t-values parametrizing the parts of the curve in quadrants I, II, and IV. Note that t = −1 is a point of discontinuity of the parametrization.

(c) Calculate dy/dx as a function of t and find the points with horizontal or vertical tangent.

Transcribed Image Text:

II -2 III 2 y I 2 IV -X