Question

What are represented by the equation $x^3+y^3+(x+y)(xy-ax-ay)=0.$

Answer 1

$x^3+y^3+(x+y)(xy-ax-ay)=0$

Factorising the equation

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

$(x+y)(x^2+y^2-ax-ay)=0$

Then, either $(x+y)=0$ or $(x^2+y^2-ax-ay)=0$

$x+y=0$ represents a straight line passing through origin.

$x^2+y^2-ax-ay=0$

represents a circle with centre $(\frac{a}{2},\frac{a}{2})$ and radius $\frac{a}{\sqrt{2}}$

So, the combined equation represents a circle and a straight line.