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# Features of a circle from its expanded equation

Sal finds the center and the radius of a circle whose equation is x^2+y^2+4x-4y-17=0, and then he graphs the circle. Created by Sal Khan.

Video transcript

We're asked to graph the circle. And they give us this somewhat
crazy looking equation. And then we could graph
it right over here. And to graph a circle, you have
to know where its center is, and you have to know
what its radius is. So let me see if
I can change that. And you have to know
what its radius is. So what we need to do
is put this in some form where we can pick out its
center and its radius. Let me get my little scratch pad
out and see if we can do that. So this is that same equation. And what I
essentially want to do is I want to complete
the square in terms of x, and complete the square in terms
of y, to put it into a form that we can recognize. So first let's take
all of the x terms. So you have x squared and
4x on the left-hand side. So I could rewrite this
as x squared plus 4x. And I'm going to put some
parentheses around here, because I'm going to
complete the square. And then I have my y terms. I'll circle those
in-- well, the red looks too much like the purple. I'll circle those in blue. y squared and negative 4y. So we have plus y
squared minus 4y. And then we have a minus 17. And I'll just do that
in a neutral color. So minus 17 is equal to 0. Now, what I want
to do is make each of these purple expressions
perfect squares. So how could I do that here? Well, this would be a perfect
square if I took half of this 4 and I squared it. So if I made this plus 4,
then this entire expression would be x plus 2 squared. And you can verify
that if you like. If you need to review on
completing the square, there's plenty of videos
on Khan Academy on that. All we did is we took
half of this coefficient and then squared it to get 4. Half of 4 is 2,
square it to get 4. And that comes straight out of
the idea if you take x plus 2 and square it, it's going
to be x squared plus twice the product of 2 and
x, plus 2 squared. Now, we can't just
willy-nilly add a 4 here. We had an equality before,
and just adding a 4, it wouldn't be equal anymore. So if we want to
maintain the equality, we have to add 4 on the
right-hand side as well. Now, let's do the same
thing for the y's. Half of this coefficient right
over here is a negative 2. If we square negative 2,
it becomes a positive 4. We can't just do that
on the left-hand side. We have to do that on the
right-hand side as well. Now, what we have in blue
becomes y minus 2 squared. And of course, we
have the minus 17. But why don't we
add 17 to both sides as well to get rid of
this minus 17 here? So let's add 17 on the left
and add 17 on the right. So on the left, we're just left
with these two expressions. And on the right, we
have 4 plus 4 plus 17. Well, that's 8 plus 17,
which is equal to 25. Now, this is a form
that we recognize. If you have the form x minus a
squared plus y minus b squared is equal to r squared, we know
that the center is at the point a, b, essentially,
the point that makes both of these equal to 0. And that the radius
is going to be r. So if we look over
here, what is our a? We have to be careful here. Our a isn't 2. Our a is negative 2. x minus
negative 2 is equal to 2. So the x-coordinate
of our center is going to be negative 2, and
the y-coordinate of our center is going to be 2. Remember, we care about the
x value that makes this 0, and the y value
that makes this 0. So the center is negative 2, 2. And this is the radius squared. So the radius is equal to 5. So let's go back to the
exercise and actually plot this. So it's negative 2, 2. So our center is negative 2, 2. So that's right over there. X is negative 2,
y is positive 2. And the radius is 5. So let's see, this
would be 1, 2, 3, 4, 5. So you have to go a little
bit wider than this. My pen is having trouble. There you go. 1, 2, 3, 4, 5. Let's check our answer. We got it right.