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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 $y=\frac{1}{x}$?

vonjd
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  • Please try to clarify your notion of distance here, the question as such is unclear. – AlexR Nov 25 '13 at 09:52
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    Related : http://math.stackexchange.com/questions/544012/why-are-two-definitions-of-ellipses-equivalent – lab bhattacharjee Nov 25 '13 at 09:53
  • @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
  • @vonjd I have read the original formulation; I did not understand your final remark here, since it made no sense to me gramatically. – AlexR Nov 25 '13 at 10:11
  • @AlexR: I am afraid I still don't understand: What part doesn't make sense gramatically?!? – vonjd Nov 25 '13 at 10:13
  • ", the distance between the two vertices". Vertices should say foci and maybe a little "which is" helps separate it from the rest; I read it as something like $|d(x,f_1) - f(x,f_2)| = 2a d(f_1, f_2)$ – AlexR Nov 25 '13 at 10:19
  • @AlexR: I edited the question - is it clearer now? – vonjd Nov 25 '13 at 10:27
  • May have a look into Article $#393$ of The elements of coordinate geometry (1895) by Loney (https://archive.org/details/elementsofcoordi00lone) – lab bhattacharjee Nov 25 '13 at 10:30
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    Related : http://math.stackexchange.com/questions/44391/foci-of-a-general-conic-equation – lab bhattacharjee Nov 25 '13 at 10:30
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    @vonjd Yes, it is. Thanks :) – AlexR Nov 25 '13 at 10:30

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