I was doing an exercise to learn about parametrization and I stumbled upon one that I thought had no answer, it asked for the parametrization of the curve formed by the intersection of $z=\sqrt(x^2+y^2)$ and $x^2+y^2-2ay=0$ and I thought it didn't have a solution as a circumference and the cone intersect in one point, then I read the solution and came to the realization that $x^2+y^2-2ay=0$ in this case corresponds to the equation of a cylinder with infinite height, as any point in the z axis satisfies the equation.
Is there a convention that says that any equation without the z axis is a 3d shape with infinite z height when the problem talks about his intersection with another equation that has the z axis (The problem didn't specify the number of dimentions)?
If that's not the case, how do you plot a cylinder and a circumference at the same time , considering both equations are the same?
Well, I think it depends on the specific problem. For instance, the equation $x^2+y^2 = 1$ represents a circle with radius 1 if you are only considering $\mathbb{R}^2$. However, when in $\mathbb{R}^3$ the equation $x^2+y^2 = 1$ represents a cylinder with radius 1 and infinite height, i.e it goes to infinity in the z-axis. For your particular problem, you are looking for the intersection of two surfaces in $\mathbb{R}^3$ so if the equation doesn't contain the z-variable then you should interpret this as infinite height in the z-axis.