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Strange Points of Locus

Date: 6/6/96 at 8:24:33
From: Anonymous
Subject: Strange Points of Locus

Given two fixed points, A and B, on a plane, if P is a moving point 
such that PA and PB are perpendicular and the locus of P is a circle, 
should we exclude points A and B?

Date: 6/14/96 at 0:54:4
From: Doctor Brian
Subject: Re: Strange Points of Locus

Yes, we should.  P can not be the same as A.  If it were, then we'd 
have the statement that AA and AB are perpendicular.

1. AA would have to be a line segment, but it's not....line segments 
have positive length, not a length of zero.

  Now, you might want to make the argument that this is a special 
case, that AA is a limiting case of the length of all perpendicular 
segments to AB through A.  The problem with that argument is that an 
identical argument can be made for all segments forming an 89 degree 
angle with AB.  Or any angle you want!  You can't really put a 
specific degree measure on the angle.

2. Another argument for this comes from the definition of cosine in 
vector spaces.  If you've ever worked with vectors, you may have run 
across the definition of the cosine of the angle between two vectors:

cos theta = (the "dot product" of the vectors)/(the product of the 
lengths of the vectors)

If one of the vectors here is AA, then its length is zero, and so the 
denominator of the expression above would be zero.  So the formula 
basically doesn't allow us to come up with an angle.

Now, you can repeat the whole thing to show why B must be eliminated 
from the locus as well.

-Doctor Brian,  The Math Forum
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Associated Topics:
High School Conic Sections/Circles
High School Geometry

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