# Rational Parametrization of a Circle & Pythagorean Triples

Today, we discuss an interesting topic rarely taught in school: The rational parametrization of a unit circle. That is, how to find the x coordinates and y coordinates of a circle expressed as a rational function? The usual parametrization of a circle is (cos t, sin t). We consider the straight line (l), passing through the point A(1,0) and the point (0,h).

The gradient of this line is $\displaystyle m=\frac{h-0}{0-1}=-h$.

The y-intercept of this line is $h$.

Hence the equation of line l is $\boxed{y=-hx+h}$.

We know the equation of the unit circle is $\boxed{x^2+y^2=1}$.

By solving the two simultaneous equations (boxed), we get a quadratic formula: $x^2(1+h^2)-2h^2x+h^2-1=0$

Solving the above using the quadratic formula gives us $\displaystyle x=\frac{h^2-1}{1+h^2}$.

Using $\boxed{y=-hx+h}$, we get $\displaystyle y=\frac{2h}{1+h^2}$.

Hence, $\displaystyle \boxed{ (\frac{h^2-1}{1+h^2},\frac{2h}{1+h^2})}$ is a parametrization of the unit circle.

We can use this to generate Pythagorean triples! Simply choose a value of h, say, $h=8$.

Then $\displaystyle x=\frac{h^2-1}{1+h^2}=\frac{63}{65}$. $\displaystyle y=\frac{2h}{1+h^2}=\frac{16}{65}$.

Substituting into $\boxed{x^2+y^2=1}$, and multiplying by the denominator, we get the Pythagorean Triple $16^2+63^2=65^2$ . Interesting?

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