The Parabola

Algebraic Definition of The Parabola

Recall that  the standard equation of the parabola is given by

 y = a(x - h)2 + k

If we are given the equation of a parabola

y = ax2 +bx + c

we can complete the square to get the parabola in standard form.

Geometry of the Parabola

We can define a parabola as follows:

 Geometric Definition of the Parabola Let F be a point on the plane and let y = -p be horizontal line called the directrix.  Then the set of points P such that FP is equal to the distance from the line to P is a parabola.

Example

Let

F = (0,2)

and

y = -2

be the directrix.  Then

FP = (x2 + (y - 2)2)1/2

and the distance from P to the directrix is given by

2 + y

Hence

2 + y =  (x2 + (y - 2)2)1/2

squaring both sides, we get

4 + 4y + y2 = (x2 + (y - 2)2) =  x2 + y2 - 4y + 4

We have

8y = x2

or
y = x2/8

In general if

y = -p

is the equation of the directrix and

V = (h,k)

is the vertex, then the Focus is at the point

F  =  (h,k + p)

and the equation of the parabola is

 y = 1/4p (x - h)2 + k

Note that vertex will always be half way between the focus and the directrix.

Example:

Find the equation of the parabola with Focus at (1,2)  and directrix y = -4.

Solution

We see that the vertex is at the point

(1, (-4+2)/2) = (1,-1)

Since the directrix is y = -4, we have

p  =  4

so that

1/4p  =  1/16

Hence the equation is

y = -1/16 (x - 1)2 - 1

Optics

Why the word focus?

Application 1:

A flashlight.
If a flashlight is to be 3 in in diameter and 2 inches deep, where should the bulb be placed?

Solution:

If the bulb is placed at the focus then the reflected light rays from the bulb will all travel in straight parallel lines outward.  We know that

y  = 1/4p x2

so that

2 = 1/4p (1.5)2

Solving gives

8p = 2.25 or p = 0.28125 inches

Exercise:  Frying an Insect

Suppose that you have a magnifying glass that is 3 inches in diameter and .5 inched deep.  How high above the ground should you hold the magnifying glass so that it burns a hole in a leaf on the ground?