Ah now I thought ellipses - definitely ellipses, I thought, but then there is the possibility of getting an astroid or hypocycloid.

That would never do, and it is so easy with the requirement being just

x to the power 2/3 plus y to the power 2/3 equal to 1

- and then you'd have ten singularities, four in the real plane, two at infinity and four complex double points - you could end up anywhere - or perhaps even worse, everywhere.

So you see, it has to be a circle.