Now
Now

that you know what the conic sections are and why they're
that you know what the conic sections are and why they're

called conic sections, let's see if we can understand the
called conic sections, let's see if we can understand the

actual equations of the conic sections a little bit better,
actual equations of the conic sections a little bit better,

and use that knowledge to be able to at least recognize them
and use that knowledge to be able to at least recognize them

when we see the equation, and then if we see the equation,
when we see the equation, and then if we see the equation,

know how to plot them.
know how to plot them.

So the first one we'll start off with is the circle.
So the first one we'll start off with is the circle.

And you've probably seen this for many years already,
And you've probably seen this for many years already,

but I'll review, if you don't know it already.
but I'll review, if you don't know it already.

The general form of an equation, or the general
The general form of an equation, or the general

equation for a circle, is x squared plus y squared is equal
equation for a circle, is x squared plus y squared is equal

to r squared, where r is the radius of the circle.
to r squared, where r is the radius of the circle.

And this would be a circle centered at the point 0,0.
And this would be a circle centered at the point 0,0.

So I'll just draw this kind of a general graph right here.
So I'll just draw this kind of a general graph right here.

So that's the x-axis, that's the y-axis, the circle will
So that's the x-axis, that's the y-axis, the circle will

look like this-- let me do it in a different color-- the
look like this-- let me do it in a different color-- the

circle would look something like-- make sure it's a
circle would look something like-- make sure it's a

circle-- well, it's supposed to be centered at 0,0, but that's
circle-- well, it's supposed to be centered at 0,0, but that's

close enough-- so its center will be right here, and the
close enough-- so its center will be right here, and the

radius if you go from the center to any point along that
radius if you go from the center to any point along that

circle will have a distance of r, so if you go from there to
circle will have a distance of r, so if you go from there to

there it's r, from there to there it's r, from there to
there it's r, from there to there it's r, from there to

there it's r, and to some degree, this formula, all it is
there it's r, and to some degree, this formula, all it is

is an extension of the distance formula, which is really just
is an extension of the distance formula, which is really just

the extension of the Pythagorean Theorem.
the extension of the Pythagorean Theorem.

So for example, the distance formula, if I want to know the
So for example, the distance formula, if I want to know the

distance between some point x,y and the point 0,0, what you do
distance between some point x,y and the point 0,0, what you do

is, you take the difference of the x's-- so x minus 0-- you
is, you take the difference of the x's-- so x minus 0-- you

square that, and then you add that to the distance between
square that, and then you add that to the distance between

the y's squared-- so that's one y point minus 0y-- y is equal
the y's squared-- so that's one y point minus 0y-- y is equal

to 0-- square that, and that is equal to the distance squared.
to 0-- square that, and that is equal to the distance squared.

So if you simplify this, x minus 0 squared, that's just
So if you simplify this, x minus 0 squared, that's just

x squared plus-- and this is just y squared is equal
x squared plus-- and this is just y squared is equal

to distance squared.
to distance squared.

So essentially, this equation is the plot of all points that
So essentially, this equation is the plot of all points that

are exactly d away, a distance of d away, from the point 0,0,
are exactly d away, a distance of d away, from the point 0,0,

and that's just a circle.
and that's just a circle.

And I'll let you think about it, I think I actually showed
And I'll let you think about it, I think I actually showed

this to you in a distance formula video, but the distance
this to you in a distance formula video, but the distance

formula just comes out of the Pythagorean Theorem.
formula just comes out of the Pythagorean Theorem.

And if that's not completely obvious for you, just think
And if that's not completely obvious for you, just think

about a little bit, and it'll hopefully become a
about a little bit, and it'll hopefully become a

little bit more obvious.
little bit more obvious.

But anyway, this was probably-- you probably already knew this,
But anyway, this was probably-- you probably already knew this,

and actually just to hit the point home, if we had the
and actually just to hit the point home, if we had the

equation like this x squared plus y squared is equal to 9,
equation like this x squared plus y squared is equal to 9,

the graph of this circle would look like this.
the graph of this circle would look like this.

So that's the x-axis, that's the y-axis, then draw the
So that's the x-axis, that's the y-axis, then draw the

circle itself, circle looks like-- close enough, and then
circle itself, circle looks like-- close enough, and then

the distance or the radius from the center of the circle,
the distance or the radius from the center of the circle,

that's going to be 3.
that's going to be 3.

There's a 9 here, why isn't the radius 9?
There's a 9 here, why isn't the radius 9?

Oh, that's because this is the radius squared.
Oh, that's because this is the radius squared.

Remember the original formula I just showed you. x squared plus
Remember the original formula I just showed you. x squared plus

y squared is equal to r squared.
y squared is equal to r squared.

So this right here is r squared, so if r squared
So this right here is r squared, so if r squared

is equal to 9, than r is equal to 3.
is equal to 9, than r is equal to 3.

It can't be minus 3.
It can't be minus 3.

I mean, it could, you can't have a negative radius, or if
I mean, it could, you can't have a negative radius, or if

you did you're just going in the other direction, but
you did you're just going in the other direction, but

it's the same thing.
it's the same thing.

So the radius is equal to 3.
So the radius is equal to 3.

So that's a circle, and that's pretty straightforward, but in
So that's a circle, and that's pretty straightforward, but in

a lot of algebra classes, they complicate the issue a little
a lot of algebra classes, they complicate the issue a little

bit by shifting the circle.
bit by shifting the circle.

So let's just shift this circle.
So let's just shift this circle.

Instead of having-- so let me just rewrite it.
Instead of having-- so let me just rewrite it.

So the unshifted circle was x squared plus y squared is equal
So the unshifted circle was x squared plus y squared is equal

to-- let me write it this way-- is equal to 3 squared, that's
to-- let me write it this way-- is equal to 3 squared, that's

the same thing as 9, and let's say that the new circle, the
the same thing as 9, and let's say that the new circle, the

shifted circle, is x minus 1 squared plus y plus 2 squared
shifted circle, is x minus 1 squared plus y plus 2 squared

is equal to 3 squared.
is equal to 3 squared.

Now all of a sudden this looks really complicated and daunting
Now all of a sudden this looks really complicated and daunting

and all the rest, but all you have to recognize is, we just
and all the rest, but all you have to recognize is, we just

substituted an x minus 1 for the-- whoops, messed
substituted an x minus 1 for the-- whoops, messed

up my pointer.
up my pointer.

We just substituted an x minus 1 for the x, and
We just substituted an x minus 1 for the x, and

we just substituted a y plus 2 for the y.
we just substituted a y plus 2 for the y.

So it has the same basic pattern as of this circle, and
So it has the same basic pattern as of this circle, and

the fact that we added or subtracted numbers from the x's
the fact that we added or subtracted numbers from the x's

and the y's tells us that we shifted the circle, and so the
and the y's tells us that we shifted the circle, and so the

next obvious question is, where did you shift it to?
next obvious question is, where did you shift it to?

And your impulse might be, oh, well maybe I shifted it to,
And your impulse might be, oh, well maybe I shifted it to,

instead of the center being 0,0, your intuition might be
instead of the center being 0,0, your intuition might be

to say that, well, now the center is at negative 1,2.
to say that, well, now the center is at negative 1,2.

And you'd be almost right, except you'd be exactly the
And you'd be almost right, except you'd be exactly the

opposite of the correct answer.
opposite of the correct answer.

The new center is now x is equal to positive 1 and
The new center is now x is equal to positive 1 and

y is equal to minus 2.
y is equal to minus 2.

And that might be unintuitive for you at first-- and you
And that might be unintuitive for you at first-- and you

might want to watch some of the videos, I think I've done them
might want to watch some of the videos, I think I've done them

already, or I've always intended to, on shifting
already, or I've always intended to, on shifting

functions-- but the way to think about it is the center
functions-- but the way to think about it is the center

here is x is equal to 0.
here is x is equal to 0.

So when x and y is equal to 0, x squared plus y squared is 0,
So when x and y is equal to 0, x squared plus y squared is 0,

you're exactly 0 away from the center, or we're at the center.
you're exactly 0 away from the center, or we're at the center.

So now if we want to be-- if we want x to be 0 away from our
So now if we want to be-- if we want x to be 0 away from our

new center, this term has to be equal to 0.
new center, this term has to be equal to 0.

And if-- just like when x was equal to 0, this term equaled
And if-- just like when x was equal to 0, this term equaled

0, so now for us to be at the center of our new circle, so to
0, so now for us to be at the center of our new circle, so to

speak, this term has to be 0.
speak, this term has to be 0.

So the new center has to be at x is equal to 1.
So the new center has to be at x is equal to 1.

Likewise, this has to be 0, and so the center
Likewise, this has to be 0, and so the center

is at y is equal to 2.
is at y is equal to 2.

Another way to think about it is this-- let's say when y,
Another way to think about it is this-- let's say when y,

you know what happens here when y is equal to 2.
you know what happens here when y is equal to 2.

Whatever part of the circle we're in when y is equal to 2.
Whatever part of the circle we're in when y is equal to 2.

There's some part of the circle we're at, in fact I could draw
There's some part of the circle we're at, in fact I could draw

it, when y is equal to 2.
it, when y is equal to 2.

Let's say this radius is 3 when y is equal to 2, we're probably
Let's say this radius is 3 when y is equal to 2, we're probably

right around there on the circle.
right around there on the circle.

We could be there or we could be there.
We could be there or we could be there.

Now we're shifting the circle, so now instead of being at 0,0,
Now we're shifting the circle, so now instead of being at 0,0,

we're going to be at 1 minus 2.
we're going to be at 1 minus 2.

So now we're going to be at the new center is x is equal to 1,
So now we're going to be at the new center is x is equal to 1,

y is equal to minus 2, the new center is there, and if I were
y is equal to minus 2, the new center is there, and if I were

to draw the new circle, it would look something like this.
to draw the new circle, it would look something like this.

I'm going to try my best to draw it still as a
I'm going to try my best to draw it still as a

circle and show you that it's been shifted.
circle and show you that it's been shifted.

No, that's not good.
No, that's not good.

Let me draw it like-- I pressed the wrong button.
Let me draw it like-- I pressed the wrong button.

That's not good.
That's not good.

Let me draw it right there.
Let me draw it right there.

That's close enough.
That's close enough.

I don't have to keep doing it.
I don't have to keep doing it.

So what we did is we shifted this circle down 2,
So what we did is we shifted this circle down 2,

and to the right 1.
and to the right 1.

So if we take its center point, we went down 2
So if we take its center point, we went down 2

and to the right 1.
and to the right 1.

And so if you think about up here, when y was equal to 2, we
And so if you think about up here, when y was equal to 2, we

could have been at this point or this point, the kind of
could have been at this point or this point, the kind of

equivalent points of the new circle are going to be here.
equivalent points of the new circle are going to be here.

Are going to be roughly here, where you're shifting it
Are going to be roughly here, where you're shifting it

down and to the right.
down and to the right.

And in order to have that same behavior in the circle
And in order to have that same behavior in the circle

there, this whole thing should be equal to 2.
there, this whole thing should be equal to 2.

So that same point on the circle, if this whole thing is
So that same point on the circle, if this whole thing is

going to be equal to 2-- because it's going to be the
going to be equal to 2-- because it's going to be the

same kind of behavior in the equation, and I hope I'm not
same kind of behavior in the equation, and I hope I'm not

confusing you there-- then the new y has to be 0, and
confusing you there-- then the new y has to be 0, and

you see it there.
you see it there.

At both of these points now, y is equal to 0.
At both of these points now, y is equal to 0.

So I know that's a little unintuitive, but I want you to
So I know that's a little unintuitive, but I want you to

sit and think about that a lot.
sit and think about that a lot.

I mean, you could just memorize it, that it's the opposite,
I mean, you could just memorize it, that it's the opposite,

when you have x minus 1 and y plus 2 that it's actually you
when you have x minus 1 and y plus 2 that it's actually you

shifted to x is equal to-- the center is now 1, minus 2, or
shifted to x is equal to-- the center is now 1, minus 2, or

you could memorize, if you like, that what makes this 0
you could memorize, if you like, that what makes this 0

and what makes this 0, and that's your new center.
and what makes this 0, and that's your new center.

But I really want you to think about it, this
But I really want you to think about it, this

is really a shift.
is really a shift.

And of course, if you were to graph it, you were to
And of course, if you were to graph it, you were to

get this thing up there.
get this thing up there.

Anyway, let me see how much time I have.
Anyway, let me see how much time I have.

Actually I didn't even keep track of the time.
Actually I didn't even keep track of the time.

I'll leave you there, I'll continue this in the next video
I'll leave you there, I'll continue this in the next video

where I'll talk a little bit about ellipses.
where I'll talk a little bit about ellipses.