In Megiddo linear programming algorithm in 2 dimensions we need to reformulate task from
\begin{array}{l} \operatorname{minimize}_{x_1, x_2}& c_1x_1+c_2x_2\\ s.t&a_{i1}x_1+a_{i1}x_2 \ge \beta_i\quad(i=1,\dots, n). \end{array}
into
\begin{array}{l} \operatorname{minimize}_{x, y} & y\\ s.t&y \ge a_ix+b_i&\quad (i\in I_1),\\ &y \le a_ix+b_i&\quad (i\in I_2),\\ &a\le x \le b \end{array}
Here i don't understand from where $a, b$ come in the last line?
Source see 2.1.
Essentially the transformation the author is doing is to rotate the LP so that the objective vector $c$ points straight down (i.e. has zero $x$ coefficient), like so:
When they do this, the coefficient of the $y$ variable is either $0$, or it is non-zero, in which case it can be scaled to equal 1. Here's an illustration of when they equal zero (and hence in which $a$ and $b$ are finite):
However, as the paper notes, we could have $a$ and $b$ equal to $\infty$, as illustrated here (the cone is supposed to be infinite):
Basically, the $a$ and $b$ just account for when the rotated LP has a constraint with zero coefficient on the $y$ variable.