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What types of symmetry do all hyperbolas have? Do all hyperbolas have rotational symmetry as well as mirror symmetry?

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You can find all that information here – Pedro Tamaroff Apr 30 '12 at 18:21
Every hyperbola has reflection symmetry in its axis and in the line through its centre perpendicular to its axis. Because it has two axes of reflection at right angles to each other, it also has $2$-fold rotational symmetry. – Brian M. Scott Apr 30 '12 at 18:23

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