Translated and non-rectangular hyperbolas

As for the ellipse, a hyperbola with axes parallel to the Cartesian axes but centre at $(x_0;y_0)$ has the form $\frac{(x-x_0)^2}{a^2} - \frac{(y-y_0)^2}{b^2} = 1$ and is recognised by completing the square starting from $Ax^2 + By^2 + Cx + Dy + E = 0,$ with $A,B$ of opposite sign (it is precisely the opposite sign of the coefficients of the quadratic terms that characterises the hyperbola, distinguishing it from the ellipse where they have the same sign).

If $a \ne b$ the hyperbola is not rectangular and the asymptotes are $y - y_0 = \pm \tfrac{b}{a}(x - x_0).$

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Completing the square · Conic · Hyperbola · Translation
Functions: Hyperbola
Skills: Analytic geometry · Using formulae