SOLUTION: Determine the center and radius of each circle and sketch graph: x^2-10x+y^2+8y=-10

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Question 103514: Determine the center and radius of each circle and sketch graph: x^2-10x+y^2+8y=-10
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2-10x%2By%5E2%2B8y=-10
The quickest way is to complete the square for both x and y. Then compare your equation to the equation of a circle and pick off the center of the circle and the radius.
%28x%5E2-10x%2BA%29=x%5E2%2B2B%2BB%5E2=%28x%2BB%29%5E2 Completing the square for x.
%28y%5E2%2B8y%2BC%29=x%5E2%2B2D%2BD%5E2=%28x%2BD%29%5E2Completing the square for y.
where 2B=-10 and A=B%5E2
and 2D=8 and C=D%5E2
Let's solve for B and A.
2B=-10
highlight%28B=-5%29
A=B%5E2
highlight%28A=25%29
Let's solve for D and C.
2D=8
highlight%28D=4%29
C=D%5E2
highlight%28C=16%29
x%5E2-10x%2By%5E2%2B8y=-10
%28x%5E2-10x%2BA%29%2B%28y%5E2%2B8y%2BC%29=-10%2BA%2BCAdd A and C to both sides.
%28x%2BB%29%5E2%2B%28y%2BD%29%5E2=-10%2BA%2BCSubstitute with the completed squares from above.
%28x-5%29%5E2%2B%28y%2B4%29%5E2=-10%2B25%2B16Substitute your calculated values for A,B,C,D.
%28x-5%29%5E2%2B%28y%2B4%29%5E2=-10%2B25%2B16
%28x-5%29%5E2%2B%28y%2B4%29%5E2=31
%28x-h%29%5E2%2B%28y-k%29%5E2=R%5E2 The equation of a circle centered at (h,k) with a radius R.
In this case,
(h,k)=(5,-4) and
R%5E2=31
R=sqrt%2831%29