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 = 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
such that FP is equal to the distance
from the line to P is a parabola.
F = (0,2)
y = -2
be the directrix.
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,
4 + 4y + y2 = (x2 + (y - 2)2) =
x2 + y2 - 4y + 4
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
Note that vertex will always be half way between the focus and the
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) =
Since the directrix is y = -4, we have
Hence the equation is
y = -1/16 (x - 1)2 - 1
Why the word focus?
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
2 = 1/4p (1.5)2
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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