A unifying definition of the conics

The definitions of parabola, ellipse and hyperbola as loci are not isolated: all three are instances of a single scheme. Let us anticipate briefly.

• Parabola: points with $d(P,F) = d(P,d)$ (one focus, one directrix).
• Ellipse: points with $d(P,F_1) + d(P,F_2) = \text{costante}$ (two foci, constant sum).
• Hyperbola: points with $\bigl|d(P,F_1) - d(P,F_2)\bigr| = \text{costante}$ (two foci, constant difference in absolute value).
• Circle: points with $d(P,C) = R$ (one centre, constant distance). It is in fact a limiting case of an ellipse with $F_1 = F_2 = C$.

The common thread: all conics are defined by distances. This is why their analytical study always passes through the formula $\sqrt{\cdots}$ and through squaring — and this is why the chapter on irrational inequalities was so important.

Links

Topics: Geometric loci
Concepts: Circle · Conic · Distance between points · Ellipse · Focus · Hyperbola · Parabola · Cartesian plane
Skills: Analytical geometry