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ralphiep
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Simple question. Is what we now call a 'rectangular hyperbola' what was once called the hyperbola of Apollonius? Thanks
What is the Hyperbola of Apollonius?
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In summary, the hyperbolas of Apollonius were named by Apollonius and were related to the conics of parabola and ellipse. The term "hyperbola of Apollonius" does not have a standalone meaning, but refers to a specific point on a conic. The hyperbolas of Apollonius are rectangular in nature, as shown in theorem 26 on page 150 of the given source.
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Infinitum
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Hi ralphiep. 
As far as I know Apollonius was the one who named the conics, i.e parabola, ellipse, and hyperbola. He related them to the well known analogy of slicing an oblique cone. That obviously includes the regular(non rectangular) hyperbolas.
As far as I know Apollonius was the one who named the conics, i.e parabola, ellipse, and hyperbola. He related them to the well known analogy of slicing an oblique cone. That obviously includes the regular(non rectangular) hyperbolas.
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tiny-tim
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hi ralphiep!
all hyperbolas of Apollonius are rectangular hyperbolas
but "hyperbola of Apollonius" on its own (i think) has no meaning …
a particular point has a hyperbola of Apollonius relative to a particular conic (and it is rectangular),
see theorem 26 (p.150) of http://books.google.co.uk/books?id=qIFzkgBikEUC&pg=PA151&dq="hyperbola+of+apollonius"&hl=en
all hyperbolas of Apollonius are rectangular hyperbolas
but "hyperbola of Apollonius" on its own (i think) has no meaning …
a particular point has a hyperbola of Apollonius relative to a particular conic (and it is rectangular),
see theorem 26 (p.150) of http://books.google.co.uk/books?id=qIFzkgBikEUC&pg=PA151&dq="hyperbola+of+apollonius"&hl=en
Related to What is the Hyperbola of Apollonius?
1. What is a Hyperbola of Apollonius?
The Hyperbola of Apollonius is a type of hyperbola that was discovered by the ancient Greek mathematician Apollonius of Perga. It is defined as the locus of points whose distances from two fixed points (foci) have a constant ratio.
2. How is the Hyperbola of Apollonius different from a regular hyperbola?
The Hyperbola of Apollonius is different from a regular hyperbola in that it has two foci, whereas a regular hyperbola only has one focus. Additionally, the Hyperbola of Apollonius has a constant ratio of distances from the foci, while a regular hyperbola has a constant difference between the distances from the foci.
3. What is the significance of the Hyperbola of Apollonius?
The Hyperbola of Apollonius is significant in mathematics because it is one of the earliest known examples of a conic section, a curve formed by the intersection of a plane and a double-napped cone. It also has practical applications in various fields such as optics, physics, and engineering.
4. How is the Hyperbola of Apollonius used in real life?
The Hyperbola of Apollonius is used in real life in various applications such as designing satellite orbits, calculating the trajectory of missiles, and determining the location of earthquakes. It is also used in optics to create a hyperbolic mirror with specific properties.
5. Can the Hyperbola of Apollonius be graphed?
Yes, the Hyperbola of Apollonius can be graphed on a Cartesian coordinate plane. The equation of a Hyperbola of Apollonius in standard form is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) is the center of the hyperbola and a and b are the semi-major and semi-minor axes, respectively. This equation can be used to plot points and draw a graph of the hyperbola.
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