In Exercises 109–111, give the center and radius of each circle. ...

Question

In Exercises 109–111, give the center and radius of each circle. x^2 + y^2 - 4x + 2y - 4 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We've got an equation of a circle and we want to figure out what is the center and the radius of this circle. So in order to do that, we're going to convert this first equation into the standard form equation of a circle. If you recall from previous lessons, that's going to be x minus H squared plus at y minus k squared equals r squared. Where h comma K. Is the center of the circle and r. Is the radius. Alright? We've got our goal in mind. We know where we're going. Now let's see what we've got to work with the to start. We've got X squared plus y squared plus six, X minus two. Y equals negative six. Okay, well let's group our axes together and our wives together. So X squared plus six X plus y squared minus two, Y equals negative six. Okay, we want these to be squared. How do we do that? Well what we're going to do here is we are going to complete the square. So first step for that, we're just going to give ourselves some more space, X squared plus six. X. Leave a bunch of space plus y squared minus two. Y. Leave a bunch of space equals negative six. And you actually leave a bunch of space after that negative +62. And that's really important. So now we've got to complete the square in order to complete the square. We're going to first look and see is there anything in front of this X term because we want to think of it as okay, we've got a X. Squared plus B. X plus C. So that we can factor it. And okay there's just one in front of that X. Squared. So we don't have to worry about that. And now we're going to look at the B. Term in order to determine our C. Term. We're going to take the B. Term and we're going to divide it by two and then we're going to square it. So for our exes are B term is six. We're going to take six divided by two. Well that'll be three and we're going to square it which will give us nine. So we're going to add nine. And what we do to one side, we're going to do to the other. Super important. Now we're going to look at our wise, is there anything in front of this Y term? Well it's just a one. Great. So we don't have to worry about the A. Term and we can just start looking at the B. Term. We've got negative two for our B. Term. We're going to take negative two divided by two. And that's going to be negative one and we're going to square it. Cool. That's going to give us one. We're going to add one after the Y. For our new C. Term there and what we do to one side we have to do to the other to keep them equal. Alright now we can start to factor so we've got our X terms, we've got X squared. Well that'll be X times X. And then they have to multiply two equal nine. Well let's do plus three and plus three and if you wanted to you could foil that really work. Just to check X times X. First x squared Outsides plus three X insides plus three X. And lasts plus nine, combine like terms X squared plus six X plus nine. That checks out. Now let's factor the wise plus, we've got y squares that'll be why times Y. And then we've got equal one. Well that could be one times one but they have to add up to be a negative two. Y. So let's do minus one times minus one to get us that minus two. Y. We will double check that real quick by foiling first. Why squared Outside minus Y insides minus Y. And last will be a plus one combine like terms Y squared minus two Y plus one. That checks out. Cool. And now bringing down that right hand side negative six plus nine plus one, that's going to give us four. So X plus three times X plus three is the same thing as X plus three squared plus y times one times Y times y y minus one times y minus one is the same thing as y minus one squared. Bring down that equals four. Cool this looks just like our standard form equation of a circle. So we've got our H values, so the center is going to be the H the H value is going to be negative three. We've got our K values, K value is going to be one. Cool, we found the center and now the radius we're going to have r squared equals four, and we take the square root of both sides and R equals plus or minus two. We don't have to worry about that minus two because it's the distance. So R is going to equal to we look at our answer choices and this is going to match answer choice. D. Well done. We'll catch you on the next one.

In Exercises 109–111, give the center and radius of each circle. x^2 + y^2 - 4x + 2y - 4 = 0

Key Concepts

Standard Form of a Circle

Completing the Square

Quadratic Terms in Circle Equations

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