Line and Unit Circle; Pythagorean TriplesDate: 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/ |
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