Question

What are represented by the equation $x^3-y^3=(y-a)(x^2-y^2).$

Answer 1

Given equation: $x^3-y^3=(y-a)(x^2-y^2)$

$\Rightarrow (x-y)(x^2+y^2+xy)=(y-a)(x-y)(x+y)$

$\Rightarrow (x^2+y^2+xy)=(xy+y^2-ax-ay)$ & $x-y=0$

$\Rightarrow x^2=-ax-ay$

$\Rightarrow x^2+ax+ay=0$

Thus we obtain 2 curves; $x-y=0$ which is a straight line and $x^2+ax+ay=0$ which is a parabola.