The Parabola

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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.

$Graph of a parabola with point P and directrix y=p and focus F$

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?

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