Coordinate systems

Parabola and rectangular hyperbola

The parabola $y^2=4ax$ is every point equidistant from the focus $(a,0)$ and the directrix $x=-a$; parametrically $(at^2,2at)$. The rectangular hyperbola $xy=c^2$ has parametric form $\left(ct,\frac ct\right)$. Parameters make tangents and normals tidy.