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I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance between the two vertices (for clarification see also this example from wikipedia).

How can I intuitively see and easily proof that this holds for $y=\frac{1}{x}$?

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Please try to clarify your notion of distance here, the question as such is unclear. –  AlexR Nov 25 '13 at 9:52
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@AlexR: What exactly is unclear to you? The first part gives a definition of the hyperbola (see also link to wikipedia which is included), the second part asks the question how to prove that this def. holds for 1/x. –  vonjd Nov 25 '13 at 10:06
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@vonjd Yes, it is. Thanks :) –  AlexR Nov 25 '13 at 10:30

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