find the center and the radius of the circle given by theequation 0 = x 2 + y 2

Question

find the center and the radius of the circle given by theequation

0 = x2 + y2 - 4x + 2y - 11
(to do this you must complete the square)

my solution

okay i just worked backwards from the given equation and luckilysomehow the numbers worked out but is there a specific way to findthe correct numbers. what is completing the square? anyways...

0 = x2 + y2 - 4x + 2y - 11
0 = (x2 - 4x + 4) +( y2 + 2y + 1) -11   >>> how i concluded this... x2 -4x...sum = -4 (x-2)2
                                                                                                     ... y2 + 2y...sum = 2 (y+1)2
(x2 - 4x + 4) +( y2 + 2y+ 1) = 11 + 4 + 1
(x2 - 4x + 4) +( y2 + 2y + 1) = 16
since (x - 2)2 + (y + 1)2 = 16 (x -x0)2 + (y - y0)2 =r2
                                                  
r2 = 16
    r = 16
    r = 4
and
(x0 , y0) = (2, -1)

this is the right answer but i think i arrived at it by fluke. sowhat is the correct way? i need a reminder of completing thesquare, a very simple example would be great.

Explanation / Answer

you are correct. seperate the terms like you did and take half of the x term andsquare it, then add to both sides. Then take half of the y term andsquare it, then add to both sides. move the constant overfirst...to clarify things. x2-4x       +y2 +2y         =11 -4/2 = -2, then (-2)2 = 4, for y , 2/2 = 1 , then12 = 1 Add to both sides x2-4x + 4     + y2 +2y +1     = 11 + 4 + 1 (x-2)2            (y+1)2            = 16

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