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.



        F = (0,2) 


        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


        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  

        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.


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


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


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?


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?


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