10.3: Parabolas
•Geometric definition: A parabola is the set of points that are equidistant from a point (called
the focus) and a fixed line (called the directrix).
•Using the geometric definition to find a formula
x
Axis of symmetry
Latus rectum
distance=px2+ (y−p)2
distance=y+p
(0, p)
(x, y)
(0,0)
y=−p(Directrix)
Note that the distance between the vertex, (0,0), and the focus and the distance between the
vertex, (0,0), and the directrix are equal. That is why focus is denoted by (0, p) and directrix
is denoted by y=−p. Now using these information we derive a closed form formula for the
above parabola:
px2+ (y−p)2=y+p
=⇒
Isolate one radical: Already done.
=⇒
Raise to Power 2: x2+ (y−p)2= (y+p)2
=⇒
Use binomial expansion and simplify: x2+y2−2py +p2=y2+ 2py +p2
=⇒
Simplify: x2= 4py
•Latus rectum: The line segment through the focus, perpendicular to axis of symmetry
with endpoints on the parabola is the Latus rectum. The length of the latus rectum is
called focal diameter. It can easily be seen that the length is 4|p|: Plug in y=pin the
the closed form formula to get x2= 4p2so x=±2pare the two end points of the Latus
rectum. Therefore, the length is 4|p|.
•Applications: The parabolic concave mirrors reflect any ray parallel to their axis of
symmetry through their focus. They are used to gather energy at their focus. Also, if you
need perfectly parallel light beams, place the light source in the focus of a concave mirror.
