Rectangular hyperbola xy=6

Statement

Show that $xy = 6$ is a rectangular hyperbola referred to its asymptotes and determine $a$.

Solution

The equation $xy=k$ is the form taken by a rectangular hyperbola $x^2-y^2=a^2$ when the reference frame is rotated by $45°$ so as to make the asymptotes coincide with the Cartesian axes. The link between the two forms is $k=\frac{a^2}{2}.$ Here $k=6$, so $xy=6$ is indeed a rectangular hyperbola referred to its asymptotes (its asymptotes are the $x$- and $y$-axes, perpendicular to each other). We find $a$: $a^2=2k=2\cdot 6=12 \quad\Longrightarrow\quad a=\sqrt{12}=\boxed{2\sqrt{3}}.$ Being rectangular, one moreover has $b=a=2\sqrt{3}$.

Links

Topics: Homographic hyperbola
Concepts: Rectangular hyperbola · Inverse proportionality
Functions: Hyperbola
Skills: Proving · Analytic geometry
Exercise type: Proof