Find the Center and Radius 6x^2+6y^2-12x+36y-36=0

Problem

6*x2+6*y2−12*x+36*y−36=0

Solution

1. Divide the entire equation by 6 to simplify the coefficients of the x2 and y2 terms to 1

x2+y2−2*x+6*y−6=0

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

(x2−2*x)+(y2+6*y)=6

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

(x2−2*x+1)+(y2+6*y+9)=6+1+9

4. Factor the perfect square trinomials and simplify the right side.

(x−1)2+(y+3)2=16

5. Identify the center (h,k) and the radius r by comparing the result to the standard form of a circle (x−h)2+(y−k)2=r2

h=1

k=−3

r2=16⇒r=4

Final Answer

Center: *(1,−3), Radius: *4

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