Hyperbola — Introduction

Definition. A hyperbola is the locus of a point in a plane that moves so that its distance from a fixed point (the focus $S$) bears a constant ratio $e$, with $e>1$, to its distance from a fixed line (the directrix $d$): $\dfrac{PS}{PM}=e$, i.e. $PS=e\cdot PM$ — again the general focus–directrix definition of section 7.1.3, now specialised to $e>1$.

Geometrically, the hyperbola results from slicing a double cone with a plane parallel to the axis (section 7.1.2, case 4) — since such a plane cuts through both nappes of the cone, the hyperbola naturally has two separate branches, one in each nappe, unlike the ellipse's single closed curve. …