Introduction

The Other Open Conic

The previous chapter defined a conic's eccentricity $e$ and showed that $e=1$ gives a parabola and $0<e<1$ gives an ellipse. This chapter completes the family: a hyperbola is the case $e>1$ — the locus of a point whose distance from a fixed focus is always a constant multiple, greater than one, of its distance from a fixed directrix. Because the point is pulled disproportionately away from the directrix as it moves outward, the curve does not close up the way an ellipse does; instead it splits into two separate, mirror-image branches that run off to infinity — the visual signature that tells a hyperbola apart from an ellipse at a glance.

What This Chapter Covers

Following the same approach used for the ellipse, this chapter places the focus and directrix conveniently relative to the coordinate axes and derives the hyperbola's standard equation, together with its parametric equations. It then identifies the hyperbola's key features — its eccentricity, foci, directrices and latus rectum — and derives the equations of the tangent and normal at a point on the curve, in both Cartesian and parametric form.