The Conjugate Hyperbola

Every hyperbola has a natural partner curve called its conjugate hyperbola: the one obtained by swapping the roles of the transverse and conjugate axes — i.e., the curve whose transverse axis is the original curve's conjugate axis, and vice versa. Algebraically, given

its conjugate hyperbola is simply the same expression with the sign of the constant flipped:

Each is the conjugate of the other — the relationship is symmetric, which is why it's called "conjugate" rather than something one-directional like "derived from."

Because $S'$ has the $y^2$ term positive, it opens up and down instead of left and right: its transverse axis lies along the $y$-axis with length $2b$, and its conjugate axis lies along the $x$-axis with length $2a$ — precisely the swap the name promises. Its own eccentricity is computed the same way as before, just with $a$ and $b$ trading places:

giving foci at $(0,\pm be')$ and directrices $y=\pm b/e'$.

A neat relationship links the two eccentricities: since $e=\sqrt{(a^2+b^2)/a^2}$ and $e'=\sqrt{(a^2+b^2)/b^2}$,

So a hyperbola and its conjugate can never both be "close to a right angle asymptote" or both be "very flat" independently — their eccentricities are locked together by this identity, which is a handy check in problems that give you one eccentricity and ask for the other.

Geometrically, if you sketch $S=0$ and $S'=0$ on the same axes, you get four hyperbola-like branches opening in the four "compass" directions (up, down, left, right) from a shared centre, all four asymptotic to the very same pair of lines found in Section 5.6 — which is exactly what the identity $S+S'=2A$ was capturing. …