Find the Center and Radius 9x^2+54x+9y^2-6y+64=0

Problem

9*x2+54*x+9*y2−6*y+64=0

Solution

1. Divide the entire equation by 9 to make the coefficients of x2 and y2 equal to 1

x2+6*x+y2−2/3*y+64/9=0

2. Rearrange the equation by grouping the x terms and y terms together and moving the constant to the other side.

(x2+6*x)+(y2−2/3*y)=−64/9

3. Complete the square for the x terms by adding (6/2)2=9 and for the y terms by adding ((−2/3)/2)2=1/9 to both sides.

(x2+6*x+9)+(y2−2/3*y+1/9)=−64/9+9+1/9

4. Simplify the right side of the equation by finding a common denominator.

−64/9+81/9+1/9=18/9

18/9=2

5. Rewrite the left side as squared binomials to match the standard form of a circle (x−h)2+(y−k)2=r2

(x+3)2+(y−1/3)2=2

6. Identify the center (h,k) and the radius r from the standard form.

h=−3

k=1/3

r2=2⇒r=√(,2)

Final Answer

Center: *(−3,1/3), Radius: √(,2)

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