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# Standard equation of a circle

Video transcript

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.