A parabola with focus and directrix

Different values of a give parabolas of different sizes, but all parabolas are the same shape (i.e. are similar to one another). The equations x =  at2, y =  2at are parametric equations for the parabola y2 = 4ax.

For a point P on the parabola, let α be the angle between the tangent at P and a line through P parallel to the axis of the parabola, and let β be the angle between the tangent at P and a line through P and the focus, as shown in the figure below;

A parabolic reflector

then α = β. This is the basis of the parabolic reflector: if a source of light is placed at the focus of a parabolic reflector, each ray of light is reflected parallel to the axis, so producing a parallel beam of light.

http://www.calculus.org/Contributions/animations.html

• Properties of the parabola with links to animated illustrations.