Canonical equation

By placing the foci symmetrically with respect to the origin on the $x$-axis, the definition of the hyperbola translates into a particularly simple equation, called the canonical one.

Property — Canonical equation

With $F_{1,2}(\mp c; 0)$, $c > a > 0$, the hyperbola has canonical form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\qquad b^2 = c^2 - a^2.$

Note the minus sign in place of the plus: it is this that distinguishes the equation of the hyperbola from that of the ellipse. The procedure to derive it is analogous to the one for the ellipse: one isolates a radical, squares, isolates the remaining radical and squares again.

Links

Topics: Homographic hyperbola
Concepts: Canonical equation · Hyperbola
Functions: Hyperbola
Methods: Canonical hyperbola
Skills: Analytic geometry