So I've drawn a circle here with radius r.
And its center is at the point, h comma k.
So its x-coordinate is h, and its y-coordinate
is k right over here.
What I want to do is figure out a general formula
for the equation of this circle.
In order to do that, we just have
to remind ourselves that a circle, all the points
on the circle, are the ones that are exactly
the radius away from the center.
Their distance from the center is exactly going to be r.
So let's think about this a little bit.
What constraints do we have to put on our x's and y's for them
to be exactly r away from the center?
So let's say that this right over here is one of the points,
so this is x comma y.
And this is one of the points that is r away,
exactly r away from the center.
So what's a relationship going to be
between x, y; h, k; and r?
And I could have put this point x, y anywhere on the circle.
I could have done it here, as well.
This would also be r away.
This is also going to be r away from the center.
Well, it might not jump out at you immediately,
but we can actually use the Pythagorean Theorem.
All we need to do, let's draw one going from the center.
Let's draw one segment that is parallel to the x-axis,
going like this.
And let's draw another segment starting at x, y,
that is parallel to the y-axis, that's completely vertical.
So this one is completely vertical.
This one is completely horizontal.
So this is going to be a right angle right over here.
So if we can figure out the-- if we
can express the height of this line in terms of the variables
we've just put out, and if we can express
the height or the width of this line right
over here in terms of these variables somehow,
then we can relate those between that
and r using the Pythagorean Theorem. r
is the hypotenuse of this right triangle.
Well, what's this height going to be?
Well, we know this point right over here
has a y-coordinate of k.
This point over here, well, that just
has a y-coordinate, I guess we could just say,
that has a y-coordinate of y.
So this distance right over here is
going to be the same exact thing as this distance
right over here.
So it's going to be y minus k.
And what's this distance going to be?
Well, same exact argument.
This is h.
And this point right over here, if we go all the way,
or this point on the x-axis, well, we'll just call that x.
So what's this distance right over here?
Well, this distance is going to be x minus h.
So what is the relationship between this side of length x
minus h, this side of length y minus k, and r?
Well, it's the Pythagorean Theorem.
The sum of the squares of the two shorter sides
is going to be equal to the square of the hypotenuse.
So one of the shorter sides has length this,
has length x minus h.
So I could write x minus h.
So the length of that side squared
plus the length of this side-- let me do this in a color
that I haven't used yet-- plus the length of this side
squared, so plus y minus k squared
is going to be equal to the hypotenuse squared,
is going to be equal to the radius squared.
And just like that, we have found the general equation
of a circle, where h comma k is the center of the circle
and r is the radius.
Any combination of x's and y's that meet this, that for which
this is true given your center, given h and k
and given the radius, then any x's and y's that
will meet these constraints are going to be on the circle.
So another way you could think about it, if someone said,
hey, look I have a center-- let me
write it this way-- the center of my circle is at the point,
I don't know, 5, negative 5.
And if they said that the radius is equal to 4,
then, we could immediately say what the equation of the circle
is going to be.
It's going to be x minus this right over here-- let me
color code this a little bit-- this right over here is our h.
This right over here is our k.
And this is our radius.
This right over here is our radius.
So the equation of the circle would be x minus h,
so x minus 5 squared plus y minus k. k is negative 5.
So if you subtract negative 5, that
would be the same thing as adding positive 5.
So y plus 5 squared is equal to r squared.
Well, r in the example I just made up is equal to 16.
So someone told you that the center is at 5, negative 5,
the radius is four, well, this is
going to be the equation of the circle.
On the other hand, if someone gave you
this as the equation of the circle,
you know that the center is going
to be, well, it's going to be at x equals 5.
And y equals, and since this is a plus,
you know that this is going to have to be, they're k.
The center is going to be at 5, negative 5.
Another way to think about is what
are the x and y's that will make each of these terms
equal to zero?
So if you have x equals 5 here, then this thing's
going to be equal to 0.
If you have y equals negative 5, then
this thing is going to be equal to 0.
If you had a circle centered at the origin at zero,
then you're just going to have x squared plus y squared is going
to be equal to the radius squared.