2025 ASRJC P1 Q4

The Folium of Descartes is a curve, defined by the equation \(x^3+y^3=3axy\), where \(a\) is a real constant. It is given that \(a \neq 0\) for this question.

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay - x^2}{y^2 - ax}\)[2]

2. The point \( \left(\frac{3a}{2}, \frac{3a}{2} \right)\) lies on this curve. Show that the equation of the normal at this point on the curve is independent of the value of \(a\).

   [2]
3. Given that the curve has a stationary point at \((pa, qa)\), where \(p\) and \(q\) are positive constants, find the exact values of \(p\) and \(q\). You need not determine the nature of this stationary point.[4]