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+8x-2y-8=0

Accepted Answer

or the following circle equation. Complete the squares. In order to write the whole equation in Standard form identified a center and radius and then graph. So standard form, a circle is given by x minus h squared plus y minus k squared equals r squared. To me to get our current equation into that form, I'm first gonna combine all of our terms and put them on the same site. So X squared plus six X for instance will be over here and y squared minus 12. Y will be over here. I'm also gonna move my 20 to the other side and say -20. Now we want to complete the square. So create the square means we'll take the B term in our quadratic or the middle term divided by two and then square it. We will add whatever result that is to the left side and the right side. So if you look here we have six X. For the X. We will take the six. First 6/2 squared is three squared which is nine Also the same for the -12 White. If you say B is in the 12th over two squared We're gonna get six squared which is 36. Now in our equation we're gonna add the nine and add 36. So we get X squared plus six X plus nine and then plus y squared minus 12, Y plus equals negative plus nine plus 36. Now we can simplify and factor all of these terms X squared plus six X plus nine. We can factor this that will turn into X plus three squared. Same for Y squared minus 12. Y plus 36. This will turn into Why -6 Squared. And finally on the right side you simplify 2 25. Now we see our center is H. K. So our H. We have a positive three here for X. That's gonna be negative three for our center And six for the K. part of our center. Now radius is just the square root of the right side. We'll square. The 25 is just five to me to find a circle with this equation X plus three squared plus five minus six. Where it goes 25 the center negative three comma six radius five. Now if you like here answer A. Has our equation, our center and our radius. I think our answers answer A. But we'll check the others just to be safe answer. Be as our credit equation. Except for the part With the radius we have a Race Square to 65 in this case which means B. Is not the correct answer. It doesn't match. Let's check see C. Has an equation but the signs are actually flipped for our center so it can't be seen. And finally D. As in craig signs for a center again and centered form. So it can't be D. Either that means the answer to our problem. This answer A. And we have a corresponding graph here on the right. Okay I hope to help you solve the problem. Thank you for watching. Goodbye.

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+8x-2y-8=0

Key Concepts

Completing the Square

Standard Form of a Circle

Graphing Circles

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