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February 6th, 2012, 12:43 AM   #1
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Hyperbola and Line intersections

Consider the hyperbola x^2 - y^2 = 1. A line in R^2 (All real numbers squared) can intersect this hyperbola in one of four ways: not at all, at one point crossing the hyperbola, at one point tangent to the hyperbola, and at two points. For each of the four cases, find a line which is an example of that case.

I've never dealt with R^2 and I don't know how to solve this equation for any of the four intersections. I don't know where to begin on this one.
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February 6th, 2012, 02:32 AM   #2
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Re: Hyperbola and Line intersections

[color=#000000]Yellow line : y=x-3 intersection at one point
Blue line: x=-1 tangent at one point
Green line: y=3/2 intersection at two points
Red line: x=1/2 intersection at zero points

[attachment=0:m6hs8ojl]hyperbola.png[/attachment:m6hs8ojl]
[/color]
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February 6th, 2012, 08:10 AM   #3
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Re: Hyperbola and Line intersections

The R^2 simply refers to two dimensional x-y space.

There are hyperboloids in 3 dimensional shape (for example electricity power station cooling towers) and indeed in more dimensions.
These and the other conic sections are all examples of quadric surfaces.
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