There are important and beautiful properties of parabolas that books do not talk about. Here are some points to help you understand parabolas.

 

The path of a thrown ball is a parabola. If students throw paper balls in class, I mention to them that the curve of the ball is a parabola.

 

When cartoons are shown on television, very often we see someone running then falling off a cliff. They draw the path as a straight line out, then a straight line down. This is wrong. It is a parabola from the moment the person left the ground.

 

Automobile headlight reflectors are parabolic. Why parabolic and not spherical? Here’s why!

 

You have learned that the equation of a parabola is y = ax2. A parabola is a quadratic equation. Let us try to understand this better by looking at the geometry. Here is how we can start from the beginning to use geometry to define what a parabola is.

 

Give a point on the x-axis, which we will call the focus. This has coordinates (a,0). Draw a line in the y direction at x = -a. This line we will call the directrix. Find points whose distance d from the directrix is equal to the distance to the focus. These points define a parabola, as we will now show.

 

The distance d from the point (x,y) to the directrix is d = a + x.

 

The distance squared to the focus (a,0) is

 

       d2 = (a – x)2 + y2

 

Solving

 

       (a + x) 2 = (a – x)22 + y2

Or

       y2 = 4ax

 

This is the equation of a parabola.

 

Using this, we will prove that light coming in from far to a parabolic reflector will converge at the focus, and, conversely, light from the focus will go out straight. This is why car headlights use parabolic reflectors. This is why satellite dishes are parabolic.

 

We explain this using geometric optics. We imagine that light travels in straight lines, called rays. We ignore other effects like diffraction, which causes blurring. When we study reflection from mirrors and refraction in lens, we use geometric optics.

 

First, we have to prove a theorem.

 

 

The tangent to a parabola makes equal angles AF and AA’.

 

Proof:

 

AF = AA’, as the distance to the focus is equal to the distance to the directrix. Therefore the triangle FAA’ is isosceles. Let T be the midpoint of FA’. The line AT is then the perpendicular bisector of the line FA’, again, this is because the triangle is isosceles.  This line AT divides the plane into two parts: one consists of points that are nearer to F than they are to A'; the other consists of points that are nearer to A'. Except for A, all points of the parabola lie in the former half.

 

Let B be any point on the parabola, and let BB’ be the line from B to the directrix. FB = BB’, as B lies on the parabola. (Recall that the distance from the focus to the directrix is the same for parabolas.)

 

SEE THE BOOK FOR MORE DETAILS.