Stationary points of an implicit cubic

A curve CC has equation x3+y3−xy=Ax^3+y^3-xy=A, where AA is a non-zero constant.

(a) Show that any stationary points on CC lie on y=3x2y=3x^2. marks

(b) Find, in terms of AA, the xx-coordinates of the stationary points, and hence determine the range of values of AA for which CC has two distinct stationary points. marks

(c) Suppose that CC has two stationary points. Determine the nature of each of the stationary points found in part (b). marks

[2023/EJC/Prelim/II/4]

Total marks: 9

Solution

If this part uses an earlier result, revisit (a) Implicit differentiation or open all working together.

(b) coordinates

Quadratics in a power

Find, in terms of AA, the xx-coordinates of the stationary points.

Hint

  1. Substitute y=3x2y=3x^2 into the curve.

  2. The result is a quadratic in x3x^3.

Answer · (b) coordinates

x±=131±1+27A3x_\pm=\frac13\sqrt[3]{1\pm\sqrt{1+27A}}.

Real cube roots; ordinate y=3x².