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>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
This is Part 8 of Ellipses.
We're going
into more advanced topics
than just simply
graphing ellipses.
We define an ellipse
and other terms associated
with ellipses such just major
and minor axes, foci,
et cetera.
And then we list the main
facts about ellipses centered
at the origin 0, 0.
We've covered the basics
of ellipses which would be
to know the equation
of ellipse in standard form
and also how
to graph an ellipse.
Now, we're going to go
on to some more advanced
topics about ellipses like,
well, what is an ellipse
by definition?
So first of all,
an ellipse is the set
of all points in the plane,
the sum of these distances
from 2 fixed points is
a constant.
I know that's a mouthful.
When you have to put things
in words, actually,
it sounds kind
of complicated sometimes.
And the fixed points are
called the foci
of the ellipse.
Now, foci is the plural
of the word focus.
So it means you have one focus
which I'll call F1
and another focus
which I'll call F2.
And it says
that to find the points
on the ellipse,
the sum of the distance
from the 2 fixed points
is constant.
So here's a way to imagine it.
Take 2 nails, okay,
and a piece on a board.
Okay, nail it in, okay?
And then take a string
that is longer
than the distance
from the 2 nails.
Okay, if it is just
between the 2 nails,
it won't work,
you won't get an ellipse.
You need something kind
of longer so you try
to pull a taut.
So imagine I have something
kind of long
that if I pull a taut,
maybe it looks something
like this for instance.
Then this would be a point
on the ellipse,
imagine that's the string,
okay?
Now, I could pull it more
taut, maybe it's over here,
so maybe this would be another
point on the ellipse.
And let's say I pull it all
the way to the right
and then back to here, right?
Maybe it's something
like that.
So if you actually imagine
pulling a taut,
pulling it around,
you get an ellipse.
So you can find some videos
on the internet
where they actually do this
with a nail
and you get something
like this, okay?
It depends
on whatever the distance is.
So you might have something
that is a lot bigger
than this,
it just depends how long
that string is.
So you think of that length
of the string as the constant.
So we're going to do it
for a particular example.
What if I have 2 foci, 4,
1 and negative--
I mean 4, 0 and negative 4,
0 which I've put
on this graph,
and a constant of 10.
Now, I'm actually showing
where some of the points
on the ellipse are already
and let me explain how I
figured that out.
If I have a constant of 10,
and I'm pulling a string taut
that's 10 units long, well,
if I pull--
start pulling from this F2
and try to pull all the way
to the right,
the furthest I can get
over it is over here at 5,
0 and that would 9 units
and then that would give me 1
unit to get back to here.
I could I pull
at the other way,
stretch it as far as possible,
I end up over here
at negative 5, 0.
So these ordered pairs are
on the ellipse, I've got 5,
0 and negative 5, 0.
The foci is somewhere inside
the ellipse.
Now, why are these points
here, 0, 3 and 0, negative 3?
Okay, so here's the reason.
If I just think
about what I know
about right triangles
and hopefully you know the
Pythagorean theorem,
notice this is a 3, 4,
5, triangle.
So this is 4 units
and this is 3 units,
then this hypotenuse would
be 5.
So imagine if I pulled
that string from F1 to F2,
I'd have 5 for this item,
5 for that side
and that's how I figured
that out.
All right, so my ellipse goes
through the 4 point, you see,
so it ends up looking
like this.
Okay, that's what an ellipse
would look
like if the foci were at 4,
0 or negative 4,
0 and if the constant,
that constant we're talking
about, ends up being 10.
So here's a picture again
and I've kind of shown
if you had these 2 links here
adding up,
I'm going to remove those now,
okay?
All right,
so you could see the ellipse
and the foci and we're going
to throw out some more
vocabulary here.
The major axis is the axis
that is on the same line
where you've got the foci.
So notice the foci are
on the X-axis, okay?
The major axis is the 2 end
points of the ellipse,
on that line.
So the major access is this
line segment A,
C. In other words,
between those ordered pairs,
negative 5, 0 and 5, 0,
all right?
So that is called the major
axis, it's a line segment.
And the vertices are the end
points of that major axis,
the negative 5,
0 and the 5, 0.
And on previous videos,
I said B and D here are also
the end-- vertices,
they actually aren't called
the vertices,
they're called the end points
of the minor axis.
The minor axis is the
other one.
It's not as long.
See how this one is longer?
It's longer going
in the horizontal direction.
So the vertices are only the
ones here,
A and C. I labeled A, B,
C and D just so it's easier
to state what the major
and minor axis was.
What's the length
of the major axis?
How far is it from A to C?
That is 10.
And what I want you
to notice here as an easy way
to get that is it's 5
to right, 5 to the left,
it ended up being just 2 times
5, okay?
All right, minor axis is B,
D. That's this other
shorter one.
So it's only
that little line segment.
And so what is the length
of that from B to D?
That would be 6 and it happens
to be, by the way, 2 times 3.
So B and D are the end points
of the minor axis
and those are what
ordered pairs?
0, 3 and 0 negative 3.
So, what I was trying
to illustrate here is
if you have 5 and negative 5
in here, that's
where I got the 2 times 5.
If you had 3 and negative 3
in here, that's
where I got the 2 times 3.
I'm looking
at these numbers here.
This is kind
of horizontal ellipse
because it is longer
in the horizontal direction.
All right,
if we look at this ellipse,
hopefully, that you understand
that the equation
of this ellipse
in standard form
since it's centered
at the origin is X squared
over 5 squared
because you notice the end
points here, 5,
0 and negative 5,
0 plus Y squared
over 3 squared equals 1.
So there's the equation
in standard form 'cause we
have taken equations
of ellipse
in the standard form
and graphed them.
And so now, when you look
at a graph,
you could also put it
in standard form.
So this is longer
in the horizontal direction,
so we call this a
horizontal ellipse.
The major axis is along the
X-axis okay?
The length
of the major axis is 10
which-- look
at the picture here,
see that 5?
That is just 2 times that 5.
The minor axis is along the
Y-axis, the shorter length
and the length of that is 6,
which is 2 times the 3.
See the smaller the number
in the bottom.
What are the vertices?
The vertices I get
from the bigger number here,
and it's on the X-axis,
will be 5,
0 and negative 5, 0.
And the end points
of the minor axis,
we don't technically call
these vertices, are--
well, they're
on the Y-axis, right?
So the smaller number is
on the Y-axis so it's 0,
3 and 0, negative 3.
And the foci
which were defined
in the first place were 4,
0 and negative 4, 0.
Now, I'm going to show you how
to get those foci.
The number
which is the important part,
the 4, comes
from this letter called C.
And C squared is going
to be just the bigger number
minus the smaller number
of the denominator.
So it ends
up being 5 squared minus 3
squared which is 25 minus 9,
which is 16.
So, think about it,
if C squared 16,
that means C is plus
or minus 4
and these are the numbers you
use to get the foci.
So now, we're going
to summarize this
for any ellipse centered
at the origin.
All right,
so here are the main facts
about an ellipse centered
at the origin.
Here's the standard
of an equation,
the standard equation
for an ellipse centered
at the origin.
So first, we're going to talk
about the horizontal ellipse
which is like the one we
just did.
It's horizontal if the number
under the X is bigger the
than the number under the Y.
So if A is bigger than B,
it's a horizontal ellipse.
So the vertices are at--
you look at the number
under the X, right?
For A squared, A,
0 and negative A,
0 and the length
of the major axis is just 2
times A. Now, we're assuming A
and B are positive numbers.
The endpoints
at the minor axis--
okay, so you take the
smaller number.
The smaller number is
under the Y squared
so it's going to be
on the Y-axis, 0,
B and 0 negative B,
and its length will be 2,
B. And how do we get the
C squared?
You just take the big number
minus the small number.
So C squared is A squared
minus B squared.
Once you figure that out,
you could figure out C, okay?
And the foci are going
to be the same,
on the same axis is
where the vertices are.
So we have A, 0, we're going
to have C, 0.
I'm not going to show you
where this comes from,
this relationship
between C squared, A squared
and B squared.
C has to do with the foci
for instance and A squared
and B squared are the
denominators
when you have this
in standard form.
But this could be shown
by using the definition
of an ellipse,
the distance formula,
the triangle inequality
and itself is quite a lot
to prove.
It can be found
in a pre-calculus book
but I'm not going to take time
to show you
where that particular
relationship comes
from in this video.
So this is--
it's a lot to take in
and we've kind
of advanced here.
This is a horizontal ellipse.
Now, if instead the number
under the Y is bigger
than number under the X,
it's very similar
but we're getting
at a horizontal ellipse.
So now, I'm writing the same
equation, okay?
And I'm going
to say B is the bigger number.
So what if B is bigger than A,
it's a vertical ellipse,
so the vertices are
on the Y-axis so we have 0,
B. Again, we're taking the
number under the Y, 0,
B and 0 negative B.
So the length
of the major axis is 2 times
B, 2 times the bigger number.
The end points
of the minor axis are at A,
0 and negative A 0,
so the length
of the minor axis is 2, A.
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