Determine if the equation x2+y2−2x+4y−4=0x^2+y^2-2x+4y-4=0x2+y2−2x+4y−4=0 is a circle, and if it is, find its center and radius.

Is a circle, center = c ( 1 , − 2 ) c\left(1,-2\right) c ( 1 , − 2 ) , radius r = 3 r=3 r = 3 .

Determine if the equation x 2 + y 2 − 2 x + 4 y − 4 = 0 x^2+y^2-2x+4y-4=0 x 2 + y 2 − 2 x + 4 y − 4 = 0 is a circle, and if it is, find its center and radius.

Hey everyone, let's see if we can solve this problem. So in this problem we're told to determine if the equation x squared plus y squared minus two x plus four y minus four equals zero is a circle, and if it is to find the center and radius of that circle. Now to solve this problem, our first step should be to see if this does take the general form of a circle. And the general form of a circle is x squared plus y squared plus ax plus by plus c equals zero. If I look at our equation, we do actually fit this form. Because we have the x squared plus y squared, we have the ax, which is negative two x, we have b y, which is four y, and then we have c, which is negative four. So as you can see, we do have the general form of a circle. So that means this is a circle that we're dealing with. Now our next step is going to be to determine the center and radius of the circle. And this step is a bit more complicated since we're going to have to convert this to standard form from general form. But once we do this, all the steps for completing the square and whatnot, you're going to be able to find the center and radius pretty quickly. So let's give this a shot. Now our first step should be to get all the x's and all of the y's combined together on one side and then get all of the constants on the other side. So I can see that we have an x squared and a minus two x. So this is going to be for ourselves, we'll have x squared minus two x. We'll combine those, then we have y squared plus four y. And then we have this negative four, which I'm gonna move to the other side and make positive. So we'll cancel the fours there, giving us that this whole thing is equal to positive four. So that's the combination step. Now what we need to do from here is complete the square. And recall for completing the square, you wanna take whatever this middle term is. We call this b. And we have a b over here as well. And then what you wanna do is take b and cut it in half and then square it. And this is going to give you the term you need to add on both sides. So doing this for this first one, we're going to have negative two over two squared. And negative two over two is equal to negative one, and then this whole thing negative one squared is going to be positive one. So that's one of the numbers we need. Now the next number we need is the for the y, so we can take this four here and cut it in half. Four over two, and we can square that. So four over two is two, and then two squared is four. So this is the other number. So what this is going to end up becoming, I'll write it over here, is x squared minus two x plus the one that we calculated, and then we're gonna have plus y squared plus four y plus, and then that number, four. And then this is all going to equal four, but we have to also add the two numbers that we calculated to make sure this equality stays the same. So it's gonna be four plus one plus four. And this is going to be what the equation looks like after doing this step. Now from here, I'm going to do two steps in one. I'm first going to combine everything on this right side, and four plus one is five, plus four is nine. And then what I'm going to do is factor each of these. Now when factoring the x's and y's, recall that what you wanna do is put it in the form x plus b over two squared, or you can put this in the form y plus b over two squared for the b associated with the y's and for the x's respectively. So if I do this for the x's, what I'm going to end up with is x plus and then whatever b over two was. Well, b over two is this inside term. We can see that that was negative one, so we're gonna end up with x minus one squared. And then this whole thing is gonna be plus y, and so we're gonna have y plus and then for b over two for the y's, it we figured out that was two. It's gonna be plus y over two squared, then that whole thing is equal to nine. Let me write this a bit better. So this is going to be the equation we end up with, that x minus one squared plus y plus two squared is equal to nine. And this is the equation in standard form. Because recall that for standard form, we have x minus h squared plus plus y minus k squared is equal to r squared, and we can see that we do have a matching form up here. So all we need to do is extract the center and radius from the standard form. Well, you can see our horizontal position is going to be one. And you can see that our vertical position well, this is a plus sign. So rather than being positive two, it's going to be a negative two. And then for our radius, well, our radius squared is going to be this nine. And if I take the square root on both sides, that will give us a radius of the square root of nine, which is three. So for our circle, we're going to end up with the center at the position one negative two, and then our radius is three. And that is the center as well as the radius of the circle. So this is how you can solve this type of problem. Let me know if you have any questions. Thanks for watching.