The Parabola

I.  Homework

II.  The Parabola

Recall that  y = (x - h)² + k

If we are given the equation of a parabola

y = ax² +bx + c

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

III.  Geometry of the Parabola

We can define a parabola as follows:

Let F be a point on the plane and let y = -p be horizontal line called the directrix.  Then the set of points P that 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 = sqrt(x² + (y - 2)²) and the distance from P to the directrix is given by

2 + y

Hence

2 + y =  sqrt(x² + (y - 2)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 (h,2k + p) and the equation of the parabola of

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,-1).  Hence the equation is

y = -(x - 1)² - 1

IV.  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

8p = 2.25 or p = .28125 inches

Application 2:  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?