line at infinity intersected with a line from points of the parabola Suppose we are looking at a parabola as a conic in $PG(2,\mathbb{K})$ with the line at infinity denoted $\ell_\infty$. I am still working on a previous problem I posted here.
Suppose we have two points $A(a,a^2)$ and $B(b,b^2)$ on the parabola. The slope of the line $AB$ is $b+a$. How does this line intersect the line at infinity? The idea of the exercise is to get the intersection, $L_{AB}$ of $AB$ and $\ell_\infty$, and then take a line from $L_{AB}$ to the origin and see where it intersects the conic. In this case I think that I always get the origin as the result, but I feel like this should not be the case. I know that the line at infinity is tangent to the parabola, does this mean that the new line will be parallel to $AB$?
 A: If you are allowed to use homogeneous projective coordinates, then
some calculations are enough to identify the objects involved.
Suppose we have two field elements $\,a,b,\,$ and let $\,c:=a+b.\,$
The point coordinates for the points $\,A,B,L_{AB}\,$ are:
$$ A = (a, a\,a, 1), \qquad
   B = (b, b\,b, 1) \qquad  \text{ and }\qquad
  L_{AB} = (1, c, 0). \tag{1}$$
The line coordinates for the lines $\,AB, \ell_{\infty}, \ell_2\,$ ($\ell_2$
is line through origin and $L_{AB}$) are:
$$ AB = (-c, 1, a\,b), \qquad
   \ell_{\infty} = (0, 0, 1) \qquad \text{ and }\qquad
  \ell_2 = (c, -1, 0). \tag{2}$$
Define the point $\,C := (c, c\,c, 1)\,$ on the parabola
and on the line $\,\ell_2.\,$
Check that all of the other incidence relations hold. For
example, line $\,AB\,$ contains point $\,A\,$ since
$$ -c(a) + 1(a\,a) + a\,b(1) = -(a+b)a + a\,a + a\,b = 0. \tag{3}$$
Projectively, this is equivalent to a circle with a point
$\,O\,$ on it regarded as the origin and then two points $\,A\,$
and $\,B\,$ on the circle determine a third $\,C\,$ so that the
line $\,AB\,$ is parallel to $\,OC.\,$ This process defines a
group operation on the circle points.
