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 - 6y -7=0

Accepted Answer

Hello Today we're going to be rewriting the given equation into the standard form of a circle and we're going to identify the center and radius of that circle. So we're going to rewrite this equation which represents a circle not located at the origin. So the standard form of that circle is going to be x minus h squared plus y minus k squared equal to r squared. So what that means is if we're going to start rewriting this into the standard form of the circle, we need to group up all of our X terms together. Group up all the white terms together and any constants that we have. We're going to move to the right hand side of the equation. So we're gonna go ahead and add 15 to both sides of this equation and once we group up all of our X terms and white terms together, the equation turns into X squared plus the quantity y squared minus 14 Y is equal to positive 15. Now we want to go ahead and get y squared minus 14, Y into the form of y minus k squared. So we're going to try to turn this into a fact Herbal trin Amiel and we can do that by completing the square And completing the square states that if we want to add a third term that turns into this into a fact herbal. Try no meal. We're going to take the second term which in this case is negative Divided by two, then square that quantity. So here negative 14 divided by two is going to give us negative seven and negative seven squared is going to give us positive 49. So we're going to be adding positive 49 as a third term two, y squared minus 14 Y. And doing this is going to give us the equation X squared plus the quantity y squared minus 14, Y plus 49. And this is going to equal to 15. Now you want to be very careful here because you want to keep the equation balanced. If you're adding 49 to the left hand side of the equation then you must also add 49 to the right hand side of the equation. So the right hand side is actually going to be 15 plus 15 plus 49 is going to evaluate to give us 64. So the right hand side of the equation is going to be 64. Now this y squared minus 14, Y plus 49 is a fact herbal try no meal as this is going to factor into why minus seven squared. So the equation that we have now is x squared plus y minus seven squared is equal to 64. And this is going to be the standard form of the equation of the circle. So we have the standard form of the equation of the circle. Now let's go ahead and identify the center and the radius and let's go ahead and start with the radius. In the standard form of the equation of the circle, the radius is represented by R squared. And in our standard form equation 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 sulfur are in order to sell for our we take the square root of both sides of the equation and taking the square root of both sides of the equation is going to give us R is equal to eight. So the radius for this circle is going to be eight. Now let's go ahead and identify the center while the center of the circle is going to be of the form H comma K. And we get our H and K values from the quantities X minus H squared and y minus K squared for the H value. We only have X squared but we can change this into x minus zero squared because x minus zero is also equal to X. So rewriting this is going to give us x minus zero squared. And this matches the form of x minus H squared. Here. Zero takes the position of H. So H is going equal to zero. Now what about K? Well, for K we're going to check the y quantity and in the problem we have y minus seven squared. Which matches the form of why minus K squared Here? seven takes the position of K. So K is going to equal to seven. So now that we have our H. Value and our K value, if we put this into the form of the center of the circle, the center is going to be zero comma seven. And that's the last piece of information we need for the equation of the circle. So just to summarize the standard form of the equation of the circle is x squared plus y minus seven squared equal to 60 for the radius is equal to eight and the center is at zero comma seven. With that 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 - 6y -7=0

Key Concepts

Completing the Square

Standard Form of a Circle

Graphing Circles

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