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>> Hi, this is Julie Harland
and I'm Your Math Gal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
This is part 10 of equations
of circles and we're going
to write the standard
in general form for each
of the circles below
that the center is
at the origin,
and there will be an ask
to graph each circle.
So we're going
to do these two problems
on this video.
So here is the first one.
Now remember, the center is
at 00, so we know the center
is at 00 and it passes
through that point.
So if you want, go ahead
and try this on your own.
Okay, let's see.
If it goes through a 00
and passes through that point,
you could imagine
approximately what
that circle looks like.
For instance,
here is the center.
It's over here.
Let's say that's negative 1,
let's say that's 1,
so we've got negative 1/2
square root of 3 over 2.
It's like, you know,
somewhere up here, right?
So, what we need to find and--
you know, that's very
approximate, right?
So I don't know exactly,
you know, where square root
of 3 over 2 is,
but it's about 1.7 over 2,
so that's about 0.85 something
like that.
So what we need to find
out is the radius, right?
Because by definition,
it's the distance from here
to here tells us the radius
of the circle.
And once you know the radius,
you could write the equation
of the circle.
So to get the radius,
we just need
to find the distance
between those two points.
So that's what will do first,
we'll find our radius.
Our radius is the square root
of-- well, we're going
to do the difference
of the X terms, right.
So there's two--
I've got a 0
and a negative 1/2,
so I'm going to write
that as 0 and negative 1/2,
I'm putting the different
of X terms, and then I'm going
to do the difference
of the Y terms, so that'll 0
and square root of 3 over 2.
Now, notice you got this whole
big square root of everything,
how about, before I go on,
if I just figure
out what are square it is,
that's what I really need
to know on top of the radius
to put in the equation,
so how if I just square both
sides, I'll get rid
of that big square root.
So we could do that.
We could just square both
sides and that makes it look a
little prettier.
We can write R squared
equals-- now,
we just get what's underneath
the square root, right,
when you square a square root.
So we've got 0 minus negative
1/2 squared, so that's going
to be 1/2 squared.
And over here,
we have 0 minus square of 3
over 2 squared, that's going
to be a negative square root
of 3 over 2 squared.
So what's 1/2 squared?
That's 1/2 times 1/2
which is 1/4.
And what's the negative square
root of 3 over 2 squared?
Well, I'm squaring negative,
so it will be positive.
Square root
of 3 times square root
of 3 will be 3,
and 2 times 2 will be 4.
So I'm going
to get 1/4 plus 3/4
which is 1.
So notice that our square is
simply 1.
So this means our circle
up here, this actually this
point is on a circle
of radius 1, okay,
going through the origin,
right, we know the radius
is 1.
So let's write the equation
of the circle now.
So, we've got--
just to keep this straight,
we know that the original was
the negative 1/2 square root
of 3-- oh no, I'm sorry.
Simpler than this.
We know that the center is
at 00, right?
So it's easy
to write this equation
in standard form, simply going
to be and it's--
remember, an equation centered
at the origin is just X
squared plus Y squared equals
R squared, so there we are.
Now that's an equation
in standard form
and in general form,
it's the same in this case,
standard and general form.
And then also said
to draw a picture throughout
the circle,
so looks like this,
that's when with just R
centered at the origin
with a radius of 1
and this is called a
unit circle.
The unit circle just means the
radius is equal to one unit.
All right,
so that's our first problem.
Now here's the second problem.
So why don't you put the video
on pause and try this one
on your own.
Again, remember,
that the center is at 00,
it passes through this point,
so there's a point
on the circle.
So remember you need
to find the radius
or R squared first,
which ever what you want
to do it.
All right, so let's do it.
So we know
that the radius will be the
distance from the center
to a point on the circle.
And this is now a point
on the circle, the square root
of 2 over 2 negative square
root of 2 over 2.
So you know,
this is somewhere, you know,
over here, right?
I don't know
where it is exactly,
but it's in the fourth
quadrant square root of 2
over 2 negative square root
over 2, and here's the center
of the circle, that's not part
of the circle.
So, that's the radius
in between those two points,
okay?
So let's just use the
distance formula.
I'm going to write R instead
of D. So again, that--
we're going
to use the square root 'cause
that's the actual formula.
We would take the difference
of the X terms
and the difference
of the Y terms.
So, let's see.
Let's put in square to 2
over 2 over 2 and 0.
Remember, it doesn't matter
if you put the X coordinate
here or the X coordinate there
first 'cause you're going
to square it.
And then for the second one,
just to be different,
I'm going to put 0 minus
and a negative square root
of 2 over 2.
Again, the order you put
doesn't make many difference.
And then to make it a little
prettier, I'm just going
to square both sides
to get rid
of that big square root sign
so that we'll figure
out what R square does.
So, we don't have the square
root anymore.
All right, so the squaring
of the square root gives you
just what's underneath the
square root
and we've got square root of 2
over 2 minus 0 squared.
So really,
we just have the square root
of 2 over 2 squared plus.
The next one is 0 minus square
root of 2 over 2.
Well, so in parenthesis,
that gives you a positive
square root of 2 over 2
and then we're going
to square it.
Now, keep in mind
that if you had switched the
order, you would have a
negative square root
of 2 over 2.
But of course
when you square it,
you're still going
to get a positive number,
that's why it doesn't matter.
So what does this give me?
So I've got to square
to numerator
and the denominator.
Square root
of 2 times square root
of 2 is 2, and 2 times 2 is 4.
So that's 2/4.
And now, this is exactly the
same thing, right?
They're both square roots of 2
over 2 squared.
So that is another 2/4
which is 4/4 which is 1.
So it' exactly
like the previous problem.
We have R squared equals 1
which of course means the
radius of the circle must
be 1.
So what's the equation
of the circle,
centered at the origin?
Remember the center was
at the origin.
Radius of 1,
X squared plus Y squared
equals 1, same exact circle.
I just gave you two different
points on the same circle.
So we've got 1 and 1,
and there we go.
So by the way,
this ordered pair, square root
of 2 over 2 negative square
roots of 2 is, you know,
somewhere around here, okay?
And on the first one,
I think I had negative 1/2
square roots of 3/4,
it was somewhere up here.
Both of them happened
to be points
on the unit circle
and the center was 00.
So, this is the standard
and general form again,
looks exactly the same.
Oh actually,
general form sometimes,
you could write minus 1
equal 0.
So, let me just clarify that.
You could write
that as X squared plus Y
squared minus 1 equals 0.
That's another way.
We usually write the
general form.
[ Silence ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.