Hyperbola and homographic function

The hyperbola is the third non-degenerate conic. Its definition is similar to that of the ellipse, but with a difference that changes everything: instead of the sum of the distances from the two foci one considers their difference (in absolute value). From this arises a two-branch curve whose branches approach two straight lines indefinitely, the asymptotes. The chapter introduces the definition as a locus and the canonical equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the characteristic elements (foci, vertices, asymptotes, eccentricity), the special case of the rectangular hyperbola and its form $xy=k$ (the old inverse proportionality), the homographic function $y=\frac{ax+b}{cx+d}$ with its asymptotes parallel to the axes, and finally the recognition of translated hyperbolas by completing the square.

Sections

• Definition
• Elements and asymptotes
• Rectangular hyperbola
• Homographic function
• Translated and non-rectangular hyperbolas

Exercises

• All the worked exercises of the chapter