What is parametrics and Parametrization of equations?

Question

I always read about parametric equations and parametrization of equations, but what is that anyway? how can I tell the difference between a Parametric equation and a 'normal' one?

Answer 1

A parametric equation basically is where variables are defined using a common parameter. For example the standard equation of a parabola can be parameterized like this:

$$y^2=4ax$$ $$y=2at$$ $$x=at^2$$ where $t$ is the common parameter. Each value of $t$ corresponds to a unique point on this parabola. For instance $t=2$ will give $(4a,4a)$.

You can then derive the equations of a tangent, or a normal to this parabola in terms of this parameter. You can also derive various relations of two points in terms of this parameter(for example, when the chord through two points passes through the focus, $t_1t_2=-1$).