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Points Equidistant from Point and Line

Date: 08/30/2001 at 08:13:33
From: Arun Yegneswaran
Subject: Locus

My teacher has given me the task of finding the locus of points that 
are equidistant from a point and a line. I have to plot as many of 
these points as possible on a sheet of paper and join them to get the 
locus. Can you please help me?

Date: 08/30/2001 at 12:47:33
From: Doctor Peterson
Subject: Re: Locus

Hi, Arun.

You will eventually learn the name and formula for this important 
curve, but for now you only have to plot some points on it.

Draw a line and a point not on the line. Now choose a distance. Set a 
compass for that distance, and draw a circle with that radius around 
the point; this will give all points the chosen distance away from the 
point. Now construct a line parallel to the given line, at the chosen 
distance. This gives all points that are that distance from the line. 
(Actually, you would need two parallel lines to make my last sentence 
true; can you see why one of the two lines is enough?)

The line and circle you have drawn will, if the distance is great 
enough, intersect in two points. At those points, the distance from 
the point is the same as the distance from the line, so they are part 
of the locus.

Repeat this for different distances, and you will have plotted the 
locus, two points at a time. You can probably find a way to do this 
faster; for instance, make a set of parallel lines spaced an inch 
apart, and then draw the circles that match them.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Equations, Graphs, Translations

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