It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$.
I want to understand this with a lot of examples.
For example, let $a,b\in K^\times$ and consider the line $L:aX+bY=Z\subset \Bbb P^2_K$.
How can I construct an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$ ?
Thank you in advance.