I'm trying to solve Exercise 3.1 in Hartshorne's Algebraic Geometry:
Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$.
I know from a previous exercise that the coordinate ring of a given affine conic $Q$ is either isomorphic to $A(Z(y-x^2))\cong k[y]$ or $A(Z(xy-1))\cong k[x,1/x]$. In the first case clearly $A(Q)\cong A(\mathbb{A}^1)$.
But how is $A(Q)\cong A(\mathbb{A}^1-\{0\})$ in the other case? Somehow I even am confused with the notation $A(\mathbb{A}^1-\{0\})$, since the only varieties in $\mathbb{A}^1$ are single points and $\mathbb{A}^1$ itself. Therefore $\mathbb{A}^1-\{0\}$ isn't an affine variety, right?
So how can I say, that $Q$ is isomorphic to $\mathbb{A}^1-\{0\}$? What is $A(\mathbb{A}^1-\{0\})$?