This example is from R. J. Walker - Algebraic Curves.
Let $S$ be the circle $x^2+y^2-y=0$ in $\mathbb{R}^2$
[it passes through $(0,0)$ and $(0,1)$; and also the horizontal lines through these points are tangent to the circle.]
If $P$ is any point on this circle other than $(0,1)$, we take projective coordinates of $P$ to be $(\rho,\rho x)$ where $\rho$ is any non-zero real number and $x$ is the abscissa of the point where line $AP$ cuts X-axis. As projective coordinates of $A$ we take $(0,\rho)$ with $\rho\neq 0$.
Question: I do not get what author want to say here. In $(\rho,\rho x$), if $\rho$ is varying over non-zero real numbers, so it is $\rho$ times $(1,x)$; now how this point $(1,x)$ is obtained from $(0,1)$, the point $P$ and the given circle?