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February 6th, 2012, 12:43 AM  #1 
Newbie Joined: Jan 2012 Posts: 29 Thanks: 0  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. 
February 6th, 2012, 02:32 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Hyperbola and Line intersections [color=#000000]Yellow line : y=x3 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] 
February 6th, 2012, 08:10 AM  #3 
Member Joined: Feb 2012 Posts: 36 Thanks: 0  Re: Hyperbola and Line intersections
The R^2 simply refers to two dimensional xy 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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