Definition:Hyperbola/Conjugate Axis

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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.