Let $K$ be a number field. I have a question about elliptic curve $Y^2Z=Z^3-XZ^2\subset \Bbb{P}_K$. I want to know what is the line which go through $[2:\sqrt{6}:1]$(This can be regarded as affine point $(2,\sqrt{6})$ in affine coordinate $Z\neq0$)) and $[0:1:0]$(infinite point).
This should be $x=2$, but I'm confused because $(2,\sqrt{6},1)$ goes through $x=2$,but $[0:1:0]$ never go through $x=2$. Where did I go wrong ?