In Exercises 53–64, complete the square and write the equation in...

Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² − x + 2y + 1 = 0

Accepted Answer

Hello Today we're going to be rewriting the given equation for a circle in standard form and we're going to use the standard form of the equation to identify the center and the radius. So what we are given is x squared plus y squared minus four X plus 18, Y plus 21 is equal to zero. Now before we start this let's just go ahead and recall the standard form of the equation of a circle that is not located at the origin. The equation of that circle is going to be x minus h squared plus why minus k squared is equal to r squared. So we're gonna try to rewrite this equation into the standard form of a circle And we're gonna go ahead and start that by grouping up our X terms together, grouping up the Y terms together and moving any constants to the other side of the equation. So let's go ahead and subtract 21 from both sides of the equation. And if we group up all of our x and white terms together, what we end up with is x squared minus four, X plus y squared plus 18, Y Is equal to -21. Now we have the groups for the X terms and the Y terms. But keep in mind that we want to convert these into factored quantities. So we're going to need to perform completing the square on both of these quantities in order to get them of the form x minus h squared and y minus k squared. And what completing the square states is if we want to add a third term to this mono meal, we're going to take the second term which in this case is negative four and we're going to divide it by two and then we're going to take that result and square it. So in order to turn this into a factory, try no meal, we're going to take negative four divided by two which gives us negative two, then we're going to square it which gives us positive for and we're gonna do the same thing two, y squared plus 18. Y. Here we're going to let the third term equal to divided by two and then squaring that quantity while divided by two is going to give us nine and nine squared is equal to 81. So now that we have the quantities that we need to make these factors will try no me als we're going to take four and add it to x squared minus four X. And we're going to take 81 add it to y squared plus 18 Y. And doing this is going to give us X squared minus four, X plus four plus the quantity y squared plus 18, Y plus 81 Is equal to -21. Now you want to be very careful here keep in mind that this equation needs to stay balanced. So if we're adding four and 81 to the left hand side of the equation, we must also added to the right hand side of the equation. So the right hand side is going to be negative 21 plus four plus 81. And negative 21 plus four plus 81 is going to simplify to give us positive 64. So the right hand side of the equation is now positive 64. Finally we can go ahead and factor X squared minus four X plus four and y squared plus 18, Y plus 81. The first quantity is going to factor 2, -2 squared. And the right quantity with Y is going to factor to why - squared. And that is going to equal to 64. And this is going to be the standard form of the equation of the circle. So now that we have the standard form of the equation of the circle, we can use this to identify the radius and center of the circle. And let's go ahead and start by identifying the radius. While the radius in the general form is represented by R squared. And here 64 takes up the spot of r squared. So we can say that r squared is equal to but we need to go ahead and sell for our in order to sulfur are we're going to take the square root of both sides of the equation and this is going to give us R is equal to eight and this is going to be the radius of the circle. Now we need to identify the center while this form of the center for a circle that's not located at the origin is going to be of the form H comma K. And we get our H comma K. From the X quantity and y quantity that has been factored. So let's go ahead and take a look at the X quantity. Here we have x minus two squared. And this matches the form of x minus H squared. But here too takes up the spot of H. So H is going to equal to two. Now let's check the y term. We have y minus nine squared and this matches the form of why minus K squared. But here nine takes up the spot of K. So K is going to equal to nine. So now we have our value of H which is two and our value of K which is nine. And if we plug this into the form of the center with the circle, not at the origin, the center is going to be at two common nine. So now we have all the piece of the information that the problem has asked us to find, we have the standard form of the equation of the circle which is x minus two squared plus y minus 10 squared equal to 64. We have the radius which is eight and we have the center which is at two common nine with all this being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² − x + 2y + 1 = 0

Key Concepts

Completing the Square

Standard Form of a Circle

Graphing Circles

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