I came across the Wikipedia page on conjugate diameters of ellipses, circles and hyperbolas, stating
[T]wo diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter.
There is a statement that the conjugate of a diameter of a rectangular hyperbola is a reflection across an asymptote. The entry further links to this page that describes it as the hyperbolic orthogonality property of two lines. Visually, it is easy and intuitive to make this deduction. However, the pages do not include any proofs - perhaps it is too lengthy. Where can I find a proof of this property?