William W. answered • 07/10/20
Experienced Tutor and Retired Engineer
This problem is a bit of a challenge because it requires you to "complete the square" twice, once for the "x" and once for the "y".
Step 1: Group the x's and y's together
(x2 + 8x ) + (y2 - 10y ) = 8
Step 2, Complete the Square. You need to find a number to add into the group of x's and another number to add into the group of y's so that each group can be written as a square. To find the number, take the number in front of the "x" which is "8", divide by 2 ("4") then square it ("16"). So add 16 to both sides of the equation, one to the group of x's and one to the other side like this:
(x2 + 8x + 16 ) + (y2 - 10y ) = 8 + 16
Now, repeat with the y's. The number in front of the "y" is "-10" and "-10" divided by 2 is "-5" and "-5" squared is 25 so add 25 to both sides of the equation like this:
(x2 + 8x + 16 ) + (y2 - 10y + 25) = 8 + 16 + 25
Now, write each grouping as a square and combine the numbers on the right:
(x + 4)2 + (y - 5)2 = 49
Step 3: Read off the center and radius from the equation
Grouped with the "x" is "4", but we must take the opposite of that, so the x-coordinate of the circle center is "-4". Grouped with the "y" is "-5", but again we must take the opposite of that, so the y-coordinate of the circle center is "5" meaning the circle center is at the point (-4, 5). The radius of the circle is the square root of the number on the right and √49 = 7 so the circle has a radius of 7
