10,11

Construct a rational parametrization of the circle $x^{2}+y^{2}=1$ by drawing lines through the point $(1,0)$.
Substitute the expression $y=t(x-1)$ into the equation of the circle. As a result, we get the equation $\left(1+t^{2}\right) x^{2}+\left(-2 t^{2}\right) x+$ $\left(1-t^{2}\right)=0$. The product of the roots of this equation is $\frac{t^{2}-1}{t^{2}+1}$, and one of the roots is equal to 1. Therefore, the other root is $\frac{t^{2}-1}{t^{2}+1}$. Further, $y=t(x-1)=\frac{-2 t}{t^{2}+1}$. In the end, we obtain the rational parametrization of the circle $\left(\frac{t^{2}-1}{t^{2}+1}, \frac{-2 t}{t^{2}+1}\right)$