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The definition of a hyperbola is

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

By this definition, the perpendicular line passing through the centre of the hyperbola, that is its conjugate axis should be called a hyperbola (may be a trivial hyperbola). As conjugate axis is set of all points such that the difference of their distance from from two foci is always $0$, which is always same, or remains constant. Is this a correct interpretation or am I missing something?

orionphy
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    Any line can be considered a "trivial" hyperbola —the common term is "degenerate"— just as any point is a degenerate circle of radius $0$, and any segment is a degenerate ellipse with minor axis $0$, etc. The conjugate of a hyperbola conveniently shares foci with the given hyperbola, but that's not really such a big deal; any symmetrically-arranged pair of points on the transverse axis can serve as foci. – Blue Jul 14 '19 at 04:00
  • @Blue That's interesting. Do you also mean that conjugate is actually defined once we have a hyperbola, and it is just that it conveniently shares the property of being a set of point at a constant difference of distance from foci. But, the same conjugate can be from that perspective a hyperbola for any set of symmetric foci's, which are infinite pairs, whereas hyperbola for a given specific foci is a unique set of curves. Can you help me to know, why the term 'degenerate' is used in this context? – orionphy Jul 14 '19 at 04:19
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    Questions about when an object is defined are a bit too metaphysical for this comment section. (Did a circle of radius $5$ with center $(12,34)$ exist before I just described it?) I'll just repeat what I wrote: Any line can be considered a [degenerate] hyperbola. ... The term "degenerate" applies to objects that live on the fringe of their definitions; typically, they have lapsed into a form that lacks the qualities we usually think of. For instance, circles of zero (or infinite) radius aren't "round". [continued] – Blue Jul 14 '19 at 05:06
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    [continuing] Often, degenerate objects are ambiguous. A "flat" triangle with collinear vertices is visually indistinguishable from a segment; you can't tell from looking what the third vertex is. Now, I may have gotten to that triangle by gradually moving the apex of an isosceles triangle toward its base; within that family, the "flat" member's "apex" is the midpoint; some other approach could have the third point coincide with a trisecting-point, or one of the vertices. Without knowledge of the process, you can't tell. [continued] – Blue Jul 14 '19 at 05:16
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    [continuing] Likewise, with your conjugate-axis hyperbola. (Note that "conjugate hyperbola" already has meaning, so don't get them confused. :) For a given hyperbola, there's an entire family —called a "pencil"— of hyperbolas sharing that hyperbola's foci; in particular, the conjugate-axis hyperbola is the member of the family with infinite eccentricity. But, if you saw that figure without the context of that original hyperbola, it's visually indistinguishable from a line, so you can't tell by looking where its foci are; any point in the plane could be one of them. [I'll stop here.] – Blue Jul 14 '19 at 05:23

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