In the following figure the plane
is slicing the cone at an angle, β,
greater than 900.
As β increases the plane will eventually become
parallel with the edge of the cone. At this point the section will be a
parabola; up to this point we have an ellipse.
The
intersection will look like the following:
An
ellipse has two axes, the major and the minor axes. Each half is
called a semi axis and are the radii of the ellipse.
There are two focal points,
always on the major axis, at the same distance from the center of the
ellipse. This distance we'll label with the lower case letter
c. The significance of these focal points will be discussed later.
The length of the x radius we'll label with
the letter a. Now, imagine sliding the
ends of the dark red line along both axis through points 1, 2 then 3.
The length of the y radius we'll label
with the letter b (shown below.)
The distance from f1 to P is also a. The triangle f1 p f2
is an isosceles triangle with a base equal to 2c
and whose legs each equal a and whose
altitude is b.
The distance from f1
to any point P on the ellipse to f2 is constant, equal to
2a. Of course, this is true for every ellipse.
The major axis can lie along the
Y axis as shown below. All previous relationships hold.
In each ellipse above (horizontal and vertical) a, b, and c form the sides of a right triangle. a
and b interchange rolls as the hypotenuse. Using Pythagorean's
Theorem, we have
b2 + c2 = a2
and a2 + c2 = b2
c2 = a2 - b2
and c2 = b2 - a2
c =
(a2 - b2)½
and c =
(b2 - a2)½
We can combine these two equations using absolute
value like so:
c = | a2 - b2 |
½
When you look at the horizontal and vertical ellipses you'll note that in
the first a is always larger than b, and in the second b is always
larger then a. We're interested in the difference of their squares,
and that must be positive, so in both cases the absolute value of this
difference will work.
The equation of an ellipse
centered at the origin is:
As with the circle, we can translate the center of the ellipse to a point (h,k)
and the like this:
and the equation in this case is :
CAUTION:
most, if not all texts will take this standard equation and interchange the a
and b to keep the 'a' with the major axis and the 'b' with the minor axis to
describe a vertical ellipse.
Like so:
I believe that doing this only adds
confusion. I chose NOT to identify a and b with these axes but, instead, with the
radii, i.e., the x radius and the y radius respectively. 'a' is
always associated with x and b is always associated with y. This is
simpler and more consistent. And it allows, without memorization,
the understanding of how a and b relate to form a vertical and a horizontal
ellipse.
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Examples: |
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1) Plot the ellipse:
a = 12, b = 3, c = | 122 - 32 | 1/2
= |144 - 9 |1/2
c = 1351/2 = 11.6
Since this ellipse is centered at the origin, the focal points
are ±11.6
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2) Plot the ellipse:
Notice the a and b values are interchanged. The larger one
determines which axis is the major axis. In this case that axis is
the Y axis. Also note that in these two examples the centers of
the ellipses are the origin. The next example is not centered at
the origin.
a=3 and b = 12, c = | b2 - a2 |1/2
= |122 - 32| 1/2 = 11.6
Note: for ellipses centered at the origin,
1) the value under the x2 term is the intersections
on the X axis, both positive and negative.
2) the value under the y2 term is the intersections
on the Y axis, both positive and negative.
CAUTION!
4 x2 + y2 = 12
IS NOT IN STANDARD FORM! We need a '1' on the right hand side
of the equation. So we divide both sides of the equation by 12
like so:
Now we have the equation in standard form and we may proceed.
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3) Plot the ellipse:
Here we have the center displaced by +1 from the origin along
the x axis and +3 from the origin along the y direction. a > b
so the major axis is parallel to the X axis. a = 5, b = 4 and
c = |a2 - b2|1/2 = | 25 - 16
|1/2 = 91/2 = 3
The new axis (dotted) has its origin at (1, 3)
We have all the information now to plot the ellipse as follows:
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4) Write the equation for the graphs to the right.
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This is another FREE ALGEBRA PRINTABLE presented to you from the
Algebra section of
K12math.com
