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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 6 of an Equation
of a Circle.
We're going
to do this problem.
Write the equation
of the circle
in standard form.
State the center and radius
of the circle.
State any X intercepts
and Y intercepts and graph it.
So we're being asked
to do a lot of things.
So the first thing we noticed
is that we don't have a circle
in standard form.
In fact, the directions are
to write to the equation
in standard form.
Remember that this is not
really in what we usually
think general form looks like.
So general form,
remember what we've wrote,
X squared plus Y squared plus
AX plus BX plus C equals 0,
and as long as not all of A,
B, and C were 0,
this would be a circle.
And so when you look
at this equation up here,
you noticed it has 3X squared
and 3Y squared.
It doesn't just have X squared
plus Y squared.
So the first thing I'm going
to point out is
if you have the same
coefficient exactly
for the X squared
and Y squared term,
you can divide both sides
of the equation
by that coefficient
to make it look
like this general form.
So for instance,
since they both are 3,
I could divide the left side
of the equation by 3 and then,
of course, the right side
as well to just get an X
squared plus Y squared plus
whatever else it happens
to be.
So even though we write the
general form like this,
you could modify it slightly
and say, well, we can have,
let's just call it another
letter, D and E--
I'm sorry,
write the same coefficient.
If these are the same
coefficient,
then actually it's really
close to general form, okay?
'Cause if you divide
everything by D,
you'd have it looked
like the first way I
expressed it.
So now, let's go ahead
and think about what
that looks
like if I divided everything
by 3.
All right,
so if I divide the left side
by 3, remember, all we have
to do is divide each term
by 3.
And we do that on both sides
of the equation
and that will give me X
squared plus Y squared plus 2X
equals 0.
That's in general form
because, remember,
as long as we have
at least one other thing
besides X squared plus Y
squared when you have 0
on the right,
then it's in general form.
So I could say that's plus 0Y
and plus 0 for instance,
that's fine.
So our next step would be
to put it in standard form.
Remember, we're trying
to get it into standard form.
Thus, we want you
to group X terms together,
the X squared--
the two X term together
because we're going
to have complete the square.
So we'll put it X squared plus
2X plus something to get
that a perfect square,
and we would want
to group the Y squared term
with the Y term.
There is no Y term
so this is awesome.
We could just leave this
as Y squared.
If you want, you could write
that as Y minus 0 Y squared,
but it's easier to just leave
that as Y squared.
What this means is the
Y-coordinate
of the center will be 0, okay?
So in other words, K is 0.
And then we have this 0
on the other side.
Now, I'm going to put a plus
of blank because whatever I
add over here on the left,
I'm going to have
to add it also on the right.
So let's go ahead
and complete the square.
So what will go in here,
we've got an X
to get the X squared
and then we've got
for our middle term plus 2X.
So we have to take half
of that coefficient.
So half of plus 2 is plus 1,
which means what you would
have to add
up here is the 1 squared or 1.
So I am adding 1
to both sides.
All right, there it is.
That's what I'm adding
to both sides.
I still have plus Y squared
and over here, I have 1,
and I now have the equation
of the circle
in standard form.
You don't have
to write Y minus 0.
It's in standard form
so just write Y squared
as long as there is no Y term.
And on the right,
you've got X plus 1,
but remember, you could write
that so that you could see the
radius, 1 is the same thing
as 1 squared.
So there we have it
in standard from,
either of these two are
standard form.
You'll see some people always
write it where you have the
something squared,
some will just simply it
and write the actual
number here.
So then, what would be--
we need to find the center
and the radius.
So what is the center, okay?
So remember if it's X plus 1,
the X-coordinate is
negative 1.
If you forget what to do,
just take what's
being squared.
In other words,
what's in parenthesis,
X plus 1, set it equal to 0
and that will tell you what
to put in for
your X-coordinate.
And for our Y-coordinate,
so this is just Y squared,
we've got 0.
And what's the radius?
The radius is 1.
So we've got our center
and our radius.
Now at this point,
you could graph it, right,
which is one
of the things you're being
asked to do.
Let's go ahead and do
that because it's going
to be easy to graph.
[ Pause ]
So there's negative 3,
it's positive 3, et cetera.
All right, so we've got
for our center,
it's over here, negative 1, 0.
I'm just going
to put a little dot here.
That's not part of the circle
and the radius is 1.
So we are going to go
from there, up 1, down 1,
to the right 1, to the left 1.
[ Pause ]
And by looking at the picture,
you can see what the X
intercepts
and the Y intercepts are.
In fact, the X intercepts are
negative 2, 0 and 0, 0.
So the X intercepts are
negative 2, 0 and 0,
0 and the Y intercept is also
just 0, 0.
It's the same point, right,
because the origin 0,
0 is on both.
So in this case,
by graphing it first,
it was really easy
to see what the X intercept
and the Y intercept was.
Now, if you--
wouldn't you've graph it
first, you can find the X
and Y intercepts by putting
in 0 for X
to find the Y intercept and 0
for Y to put an X intercept.
So let's go ahead
and take our original equation
and do that.
So here is our original
equation, 3X squared plus 3Y
squared plus 6X equals 0.
We could plug it in there,
but it's actually easier
to divide by 3
and then go ahead and plug it
into this modified equation
after we divide it by 3.
And so, let's plug X equals 0
and 0, 1 for X to figure
out the Y intercepts, right?
For the Y intercept,
you should have X equals 0.
So that will give you 0
squared plus Y squared plus 2
times 0 equals 0.
Well, that's just going
to give you Y squared is 0
and taking the square root
of both sides is just 0.
There is no plus or minus 0.
So the Y intercept,
when you put in 0 for X,
you got 0 for Y. Now
if you want
to get the X intercepts,
you're going to put in 0 for Y
and do the same thing.
Same equation, right?
So we've got X squared plus--
we're putting in 0
for Y this time, 0 squared.
Let's see, plus 2X equals 0.
So this gives me X squared
plus 2X equals 0
and then we factor.
Remember, when you have an X
squared term, you always want
to set it equal to 0
and factor.
So either X equals 0
or X equals 2--
I'm sorry,
X equals negative 2.
So those will give me--
when you put in 0 for Y,
X could be 0 or when you put
in 0 for Y,
X could be a negative 2.
Now, we should get the
same exact.
So this is an algebraic way
of doing it.
We're going to get the same X
and Y intercepts,
but of course, it was easier
to graph it first
and just visually note
where they were.
[ Pause ]
All right, so we did it all.
We put it in standard form,
we found the center,
we found the radius,
we stated the X intercepts,
the Y intercept,
and we drew the graph.
And if I just say,
what are the intercepts?
You will just say negative 2,
0 and 0, 0, right?
I just like listed them
so you would--
it's not like there are three
different intercepts, right?
'Cause I have 0,
0 for both of these.
So there is another example
for you.
[ Pause ]
We're giving something that's
not quite looking
like general form
but close enough.
Once you divide it by 3,
it looks exactly
like general form
because you completed the
square and we're able
to answer all
of those questions.
[ Pause ]
And finish the problem.
Okay. Now,
we haven't done any problems
where I've given you
like the diameter
or where I've given you some
other point on the circle.
There's a lot
of other tricky questions you
could be asked about how
to get the correct equation
of a circle and so,
look to the next videos
if you want
to see some trickier types
of problems.
[ Pause ]
Please visit my website
at yourmathgal dot com
where you can view all
of my videos
which are organized by topic.