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  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 $x^2 + y^2=36$

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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.

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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
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