how can I distinguish between these curves

Question

the curve defined as $$f(x,y)=c$$ and $$f(x)=c$$ are plane or space curve i am confuse because $x^2 + y^2=4$ and $x^2=5 $ are both plane curves.But how can i decide about $f(x,y)=c$ and $f(x)=c$ whether they are plane or space curve. thanks in advance

Answer 1

You can't, from the formula alone, tell whether there exist other letters that just happen to not appear. In the plane $x^2 + y^2 = 1$ gives a circle, while in space it gives an infinitely tall cylinder, and just from "$x^2 + y^2 = 1$" there is no way you can tell which one it is. You just have to take it from context.