Definition:Hyperbola/Conjugate Axis
Definition
Consider a hyperbola $K$ whose foci are $F_1$ and $F_2$.
Let $PQ$ and $RS$ be line segments constructed through the vertices of $K$ parallel to the minor axis of $K$ and intersecting the asymptotes of $K$ at $P$, $Q$, $R$ and $S$ as above.
Construct the line segments $PR$ and $QS$.
Let $C_1$ and $C_2$ be the points of intersection of $PR$ and $QS$ with the minor axis of $K$.
The conjugate axis of $K$ is the line segment $C_1 C_2$.
Also defined as
Some sources do not consider the minor axis of a hyperbola separately from the conjugate axis, and instead define the conjugate axis as the infinite straight line that coincides with it.
From D.M.Y. Sommerville: Analytical Conics (3rd ed.):
- There is no minor axis, but the other axis of symmetry, the $y$-axis, is called the conjugate axis.
However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we prefer to keep the concepts separate.
Some sources use the term conjugate axis to mean any arbitrary line segment on the minor axis.
Also see
Linguistic Note
The plural of axis is axes, which is pronounced ax-eez not ax-iz.
Compare basis.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {V}$. The Hyperbola: $2$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conjugate axis
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbola
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjugate axis
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbola
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate axis
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbola