Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
  1. construct two lines through the point $(3,1)$ that are parallel to the line $x=7$

  2. construct two lines through the point $(3,1)$ that are parallel to the line $x^2 + y^2=36$

share|improve this question
1  
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find rude writing a question that looks like it consists only of a verbatim quote; please consider rewriting your post. –  A.P. May 6 '13 at 20:05
add comment

1 Answer

So you are using the upper half plane model by Poincaré, right? In that case, a hyperbolic line is a line or circle perpendicular to the real axis. And two lines are parallel if they do not intersect (unless your definition of “parallel” is lines meeting at an ideal point, see below for that). So one of many possible constructions would look like this:

Fig. 1: ultraparallel lines

The two small circles are parallel to both the vertical line for the first question, and the big circle for the second question. They also both pass through $(3,1)$.

If your definition of “parallel lines” is not the lack of a point of intersection, but instead an ideal point of intersection at infinity, then the picture would look more like this:

Fig. 2: limit parallel lines

You can construct these easily if you know how to construct a ciorcle through three points, since all circles will pass through the point $(3,-1)$ as well.

Wikipedia calls this “asymptotic”, as opposed to “ultraparallel” for the above situation. Whether you define ”parallel” to mean “asymptotic” or “ultraparallel” or both depends on context, but personally I prefer to use the term to denote both alternatives together.

share|improve this answer
    
At least in my experience it's not uncommon to have 'parallel' mean lines meeting at a point at infinity, and to use 'ultraparallel' for the sort of line you describe; I would certainly take the question in that spirit, though without more context it's impossible to say for sure. –  Steven Stadnicki May 6 '13 at 22:57
    
@StevenStadnicki: Augmented my answer to cater for that definition as well. –  MvG May 6 '13 at 23:13
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.