A general philosophy of the subject is that it is fruitful to understand how a conic Q intersects lines. It is not just the intersections of Q with a single line which are significant, but its intersections with pencils of lines. In this chapter we pursue this philosophy in the special case of a parallel pencil of lines. Any line in the pencil intersecting Q twice determines a chord with a unique midpoint. Section 7.1 establishes that the midpoints lie on a line, at least provided the delta invariant is non-zero. In the next section we use this ‘midpoint locus’ to introduce axes of symmetry, a significant visual feature of a conic. For instance ellipses and hyperbolas have two perpendicular axes through their centre, whilst a parabola has a single axis. Moreover, for line-pairs the axes are the perpendicular bisectors of the component lines, familiar from Chapter 6. The final section illustrates an exceptional situation, namely parallel pencils whose general line meets Q in a single point: that leads us to the concept of ‘asymptotic directions’ for a conic, and the associated classical idea of an ‘asymptote’.

Midpoint Loci

Consider the intersection of a conic Q with the parallel pencil of lines in a general direction (X, Y). This section revolves around the fact that the midpoints of the resulting family of parallel chords lie on a line, the ‘midpoint locus’ associated to the direction (X, Y).