In ordinary analytic geometry, the equation of a conic is Axx + Bxy + Cyy + Dx + Ey + F = 0, where A, B, C, D, E, and F are reals. In the homogenous coordinates of affine geometry, this equation becomes Axx + Bxy + Cyy + Dxz + Eyz + Fzz = 0As usual, we take z=0 as the line at infinity. Then the conic is a parabola if BB-4AC=0, is an ellipse if BB-4AC< 0, and is a hyperbola if BB-4AC> 0.
Consider an ellipse. When z=0, we have Axx + Bxy + Cyy = 0. But because BB-4AC< 0, this has no solutions; the ellipse does not intersect the line at infinity.
Finding the intersections of a parabola with the line at infinity amounts to solving the quadratic Axx + Bxy + Cyy = 0 under the condition BB-4AC=0. This has the single solution x/y=-B/2A; the square root term in the quadratic formula vanishes by the given condition. This solution gives us the single point (-B, 2A, 0) on both the parabola and the line at infinity, and we can verify that this is the direction of the axis of the parabola.
We can use similar arguments to show that the hyperbola intersects the line at infinity in two places. Since the square roots aren't particularly illumunating, we'll look at two easy examples:
The hyperbola xy=1 (in Cartesian coordinates) has Cartesian asymptotes x=0 and y=0. In homogenous coordinates, this hyperbola is xy=zz, and includes two points at infinity corresponding to the two asymptotes: (0,1,0) and (1,0,0). Similarly, the hyperbola yy-xx=zz includes the points at infinity (1,1,0) and (1,-1,0). These points correspond to the 45-degree lines of Cartesian coordinates, which are the asymptotes of the corresponding hyperbola yy-xx=1.