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Line and Unit Circle; Pythagorean Triples


Date: 04/16/2001 at 04:53:30
From: Bonny Kennedy
Subject: The Unit Circle, x^2 + y^2 = 1

I am lost.

Let (X,Y) be a point in the first quadrant on the unit circle 
x^2 + y^2 = 1, and let m be the slope of the line passing through 
(X,Y) and the point (0,-1). Express the coordinates (X,Y) in terms of 
m and explain how this representation can be used to generate 
Pythagorean triples (three non-negative integers that satisfy the a^2 
+ b^2 = c^2).


Date: 04/16/2001 at 13:06:59
From: Doctor Rob
Subject: Re: The Unit Circle, x^2 + y^2 = 1

The line with slope m through (0,-1) is:

     y + 1 = m*x
or
     y = m*x - 1

You want to find the points of intersection of this line with the 
circle x^2 + y^2 = 1. Do this by substituting for y from the above 
equation, and solving the resulting quadratic equation. Of course you 
know that x = 0 will be one solution. It is the other solution you 
seek. Then x will be expressed as a function of m. Substitute this 
into the above equation to get y as a function of m, too.

To get Pythagorean triples, let m be a rational fraction greater than 
1, that is m = r/s for some integers r > s > 0. You might as well 
assume that it is reduced to lowest terms, so that r and s have no 
common factor bigger than 1. Substitute that into the equations for x 
and y in terms of m, and simplify. Let c be the denominator of both 
these expressions, and a and b the numerators. Then a^2 + b^2 = c^2, 
with a, b and c positive integers.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Geometry
High School Linear Equations
High School Number Theory

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