lines passing through two points in $\mathbb{P}^n$

Question

The reference is the Example 7.5. b) from ag notes by Gathmann. The paragraph explains the idea of projection from a point. He claims that the unique line passing through $a=(1:0:\cdots:0)\in\mathbb{P}^n$ and $x\in\mathbb{P}^n-\{a\}$ can be parametrized by $$P:=\{(s:tx_1:tx_2:\cdots:tx_n)|(s:t)\in\mathbb{P}^1\}$$ However, I think, the point $a$ would correspond to an actual line in $\mathbb{A}^{n+1}$ as below $$L_a=\{(s,0,\cdots,0)\in\mathbb{A}^{n+1}|s\in k\}$$ and the point $x$ corresponds to the line $$L_x=\{(tx_0,\cdots,tx_n)|t\in k\}$$ Hence the "line passing through them in $\mathbb{P}^n$" should be something like a plane, or a linear combination of these two lines in $\mathbb{A}^{n+1}$, i.e. $$P_{a,x}=\{(s+tx_0,tx_1,\cdots,tx_n)\in\mathbb{A}^{n+1}|s,t\in k\}$$ Then I do not see how do we get $P$ from $P_{a,x}$ by simply projectivization. The $tx_0$ part disappears. Any help is appreciated. Thank you very much!

Answer 1

With the helps and suggestions from comments, I post my own answer here: We scale $L_a$ so that $L_a=\{(s+tx_0:0:\cdots:0)|s,t\in k\}$. Then by looking at the linear combination of the new $L_a$ and $L_x$, we are able to solve it.