Forming a generic line between two points of a sphere

Question

Consider the unity sphere $S^1=\{(x,y) \in \mathbb{R}^2 \, | \, x^2+y^2=1\}$. I want to show that if $p,q \in S^1$ then $S^1 \setminus \{p,q\}$ is not connected.

My idea is to take the line formed by $p$ and $q$ and showing that the part over and the part under the line forms a non-trivial spliting of $S^1$. But my question is:

Given two generic points $p,q \in S^1$, how can I define a generic equation of that line?

Answer 1

I suggest that you use the fact that $S^1\setminus\{p\}$ is homeomorphic to $\Bbb R$. It follows that $S^1\setminus\{p,q\}$ is homeomorphic to $\Bbb R$ minus one point. And $\Bbb R$ minus one point is disconnected.

Answer 2

The implicit equation of a line by two points is

$$\vec{ps}\times\vec{pq}=0$$

where $s$ is any point on the line.