The Parabola
Algebraic Definition of The Parabola
Recall that the standard equation of the parabola is given by
If we are given the equation of a parabola
y = ax^{2} +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 = (x^{2} + (y  2)^{2})^{1/2}
and the distance from P to
the directrix is given by
2 + y
Hence
2 + y = (x^{2} + (y  2)^{2})^{1/2}
squaring both sides,
we get
4 + 4y + y^{2} = (x^{2} + (y  2)^{2}) =
x^{2} + y^{2}  4y + 4
We have
8y = x^{2}
or
y = x^{2}/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
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 x^{2
}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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