>> 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 1 of Equations
and Graphs of Ellipses.
We're going to talk
about ellipses centered
at the origin.
In this video, we do--
we graph the following ellipse
that has this equation.
All right, we're going to go
over the basics of ellipses,
not everything about ellipses
and how they're defined
necessarily but just
about their equations
and their graphs.
So the equation
of an ellipse centered
at the origin is at the form
that you see right here,
X squared over A squared plus
Y squared over B squared
equals 1.
Now, A and B are just numbers
and they can't be 0.
Okay, so that's one
important thing.
So let's just start off
with what if A
and B were the same number?
So let's see what would happen
if I had written
this equation.
How would I graph X squared
over 2 squared plus Y squared
over 2 squared equals 1
for instance?
Well, really,
that's the same thing
as saying X squared
over 4 plus Y squared
over 4 equals 1.
And then you might notice
that if you multiply both
sides by the least common
denominator of 4,
in other words,
each term gets multiplied
by 4, 4 times the first term
is X squared,
4 times the 2nd term is Y
squared and 4 times 1 is 4.
And that you might recognize
as the equation of a circle.
[ Pause ]
So the first thing you might
think is wait a minute,
what happened here?
I was talking about an ellipse
and I get an equation
of a circle.
This is the equation
of a circle
with radius 2, right?
Because 4 is 2 squared
so if I were going
to graph it,
we know the center is 0, 0,
we would just go 2 spaces
in each direction
and we would have this circle.
So the first thing I want
to you to realize is there's
something about this equation
of ellipse and the equation
of a circle that's kind
of similar, all right?
The difference is when A
and B are different numbers,
what's going to go on?
So if these are really the
same number and you put it
in this form, hopefully,
you see that you could
directly write
that as an equation
of a circle and graph it.
So we want to see what happens
if these are different
numbers, so let's try that.
All right,
so how would we graph X
squared over 25 plus Y squared
over 9 equals 1?
The first thing I want to you
to notice is our little trick
of just multiplying
by the least common
denominator will not give us
a circle.
The least common denominator
is 25 times 9
and you would get a different
coefficient for your X and Y,
so this is not going
to be a circle.
So let's see.
To graph something,
if you don't know what it
looks like,
what you could do is just make
a table of ordered pairs
and one simple thing
to do is think
about what the X
and Y intercepts are.
So how would we get the
X intercepts?
Remember, you're going
to let Y equals 0
in the equation.
So if I put in 0 for Y
in this equation,
I'll get X squared over 25.
This can be plus 0, right,
if I put in 0 squared?
So this term basically is
going to go away.
I'll have X squared
over 25 equals 1.
Now, we could multiply both
sides by 25
and that will give us X
squared equals 25.
And then we could take the
square roots of both sides
so that we get X as plus
or minus 5.
At least that's one way
to solve this equation,
X squared equals 25.
You could also subtract 25
from both sides, factor it,
set each factor to equal to 0,
you're still going
to get X is plus or minus 5.
So what are X intercepts going
to be?
Well, when I put in 0 for Y,
X could be 5
or X could be negative 5.
So I have 5,
0 and I have negative 5, 0.
So those will be our
X intercepts.
Now, why don't you put the
video on pause and try
to figure out what the Y
intercepts are?
You're going
to do a similar thing.
You want to get the Y
intercepts so you're going
to end this equation up here.
You're getting
to let X equals 0
and then figure
out what the Y intercepts are.
Right, did you do it?
Okay, so if you did this,
if you let X equals 0,
what's going
to happen is this first term,
X squared over 25,
will pretty much disappear
and you'll get Y squared
over 9 equals 1.
And then when you multiply
both sides by 9,
you'll get Y squared as 9.
This time,
you should have got Y equals
plus or minus 3.
So your Y intercepts are--
well, you let X equals 0
so with 0, 3 and the other one
with 0 negative 3.
So look back up here
at our original equation,
we found 4 ordered pairs,
the X and Y intercepts
for this equation,
X squared over 25 plus Y
squared over 9 equals 1.
Let's graph those 4
ordered pairs.
All right,
so here's our equation,
here are ordered pairs I just
put in the table, X and Y
and so let's graph this.
We have 5, 0,
that will be right here,
and we have negative 5, 0.
We've got 0, 3 and we have 0,
negative 3.
Well, that's 4 ordered pairs
so that really doesn't
to tell us exactly what this
is going to look like
but you could plug
in some other values for X
and Y and see what you get.
So let's plug
in some more values like,
I wonder what it is when,
you know, X is, you know, 3.
I wonder what happens
if you plug in 3 for X.
So I go back
up to this original equation,
I could put in 3 for X
so that gives you 9/25 plus Y
squared over 9 equals 1.
All right, so we would have
to solve this equation
so I could subtract 9/25
from both sides
or I can multiply both sides
by the least common
denominator, lots of ways to--
actually, I'm going to do
that on the next page-- oops.
All right,
so we have this equation,
so how could we solve this?
How about if I just subtract
9/25 from both sides?
I have Y squared over 9.
Now, 1 minus 9/25 is the same
thing as 25/25 minus 9/25.
So I have Y squared
over 9 equals 16/25
and then I can multiply both
sides by 9 so that I have 2
and let's say--
sorry, that's 144,
16 times 9 is 144/25.
So Y, taking the square root
of both sides,
this is square root
of 144 is 12 and square root
of 25 is 5
so that's the same thing
as plus or minus 2 and 2/5.
All right, so remember,
I plugged in X equals 3,
right?
And I got 2
and 2/5 plus or minus.
So let's go back
and plug that in.
So I got 2 and 2/5
and I also got negative 2
and 2/5 when I plugged in 3.
So let's see, if I go over 3,
it says go up 2 and 2/5.
All right, it's up right here
and there is this other one
down here.
As it turns out when you pug
in negative 3,
the same thing will happen
because when you square the X
up here, everything will come
out the same.
So I'm also going
to get this ordered pairs
over here, negative 3,
2 and 2/5, et cetera.
And you could do a whole bunch
of ordered pairs
so you get enough dots,
but what's going
to happen is you get actually
of this-- oops,
squished circle.
Let's try graphing this again.
All right,
and this is what an ellipse
looks like,
it's kind of an oval, right?
So when these denominators
here in this form are
different numbers,
it has an oval shape,
it's not going
to be a circle, okay?
And usually, a circle is kind
of especial looking ellipse.
Usually, when we talk
about an ellipse,
we mean that it will look oval
looking most
of the time, right?
All right, so,
can you see an easy way
to maybe get this
without putting
in all this ordered pairs?
Let's regroup here.
So we look
at our original equation,
right?
X squared over 25 plus Y
squared over 9 equals 1,
and this is
in standard form, okay?
That's a standard form
of an ellipse and it's
because it actually helps us
figure out how
to draw the ellipse.
Here's the idea,
standard form is often written
so that you could take
that number
in each denominator and write
as something squared
so that you can think of that
as X squared
over 5 squared plus Y squared
over 3 squared equals 1.
And notice
that the X intercepts
are what?
The X intercepts were plus
or minus 5,
and the Y intercepts were plus
or minus 3.
See that denominator here,
the 5 squared?
That tells you
where on the X axis
to put the intercepts,
5 and negative 5,
and then the denominator
into the Y,
since the 3 squared will plus
or minus 3.
And that's the simple way
to draw the ellipse,
you'll put those 4 ordered
pairs, don't worry
about all these ones
in the middle
and you're just sketching it,
right?
If you want
to get a really good sketch,
you would have
to like try a lot
of other numbers
like we figured out 3,
2 and 2/5.
And, you know,
if you could plug in all sorts
of ordered pairs
and get a whole bunch
of little dots,
so you might get a little bit
nicer looking ellipse,
that's what a graphing
calculator would do for you.
So that's how we graph an
ellipse of this form.
So if it's centered
at the origin and we have it
in this form,
it's easy to see what the X
and Y intercepts are.
So let's write this
out in a nice way.
All right, let's summarize.
An ellipse centered
at the origin
in standard form is written
as, X squared
over A squared plus Y squared
over B squared equals 1.
Now, sometimes,
if this is like 3 squared,
you'll just see people write
the number 9.
That really doesn't matter,
it's up to you whether you
want to write it
as 3 squared or 9.
The X intercepts are A, 0,
and negative A, 0.
In other words,
they cross the X axis at A
and negative A. The Y
intercepts are 0, B and 0,
negative B
which means crosses the Y axis
at B and negative B.
And these points here are
called the vertices
of the ellipse.
And just note, when A
and B are the same,
it's really just a circle
of radius A
which of course is the same
thing as radius B since A
and B are the same thing.
So we are only talking
about an ellipse centered
at the origin right now,
and we've only done one
in standard form.
And I'm not talking about how,
you know, although there's a
lot more to ellipses
than this, this is just sort
of basics of understanding
what the equation
of an ellipse looks like
and how you would graph them,
write the intercepts
which are the vertices
in this case.
So we'll do a few more
problems, also,
descent problems
that aren't already
in this form so we'll figure
out how to put it
in standard form, et cetera,
in the next videos.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.