Why does $ x^2+y^2=r^2 $ have uncountably many real solutions? What is exactly the reason the equation of a cirle of radius $ r $ and centered at the origin has uncountably many solutions in $\mathbb { R} $?

The mapping $f\colon\mathbb{R}\to\mathbb{R}^2$ defined by
$$
f(t)=\left(r\frac{1-t^2}{1+t^2},r\frac{2t}{1+t^2}\right)
$$
is injective and its image is the circle with center at the origin and radius $r$, except for the point $(-r,0)$.
