Give the center and radius of the circle represented by the equation. \[ x^{2}+y^{2}-12 x+6 y=-44 \] The center of the circle is \( \square \). (Type an ordered pair, using integers or fractions.) The radius of the circle is \( \square \). (Type an integer or a simplified fraction.)

To find the center and radius of the circle represented by the equation \[ x^{2}+y^{2}-12 x+6 y=-44, \] we first need to rewrite the equation in standard form. The standard form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2, \] where \((h, k)\) is the center and \(r\) is the radius. ### Step 1: Rearranging the equation We start by moving \(-44\) to the other side: \[ x^{2} + y^{2} - 12x + 6y + 44 = 0. \] ### Step 2: Completing the square Next, we will complete the square for the \(x\) and \(y\) terms. #### Completing the square for \(x\): 1. Take the coefficient of \(x\) (which is \(-12\)), halve it to get \(-6\), and square it to get \(36\). 2. Rewrite the \(x\) terms: \[ x^{2} - 12x = (x - 6)^{2} - 36. \] #### Completing the square for \(y\): 1. Take the coefficient of \(y\) (which is \(6\)), halve it to get \(3\), and square it to get \(9\). 2. Rewrite the \(y\) terms: \[ y^{2} + 6y = (y + 3)^{2} - 9. \] ### Step 3: Substitute back into the equation Now substitute these completed squares back into the equation: \[ ((x - 6)^{2} - 36) + ((y + 3)^{2} - 9) + 44 = 0. \] This simplifies to: \[ (x - 6)^{2} + (y + 3)^{2} - 36 - 9 + 44 = 0, \] which simplifies further to: \[ (x - 6)^{2} + (y + 3)^{2} - 1 = 0. \] ### Step 4: Rearranging to standard form Now, we can rearrange this to standard form: \[ (x - 6)^{2} + (y + 3)^{2} = 1. \] ### Step 5: Identifying the center and radius From the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\): - The center \((h, k)\) is \((6, -3)\). - The radius \(r\) is \(\sqrt{1} = 1\). ### Final Answer The center of the circle is \((6, -3)\) and the radius is \(1\). - The center of the circle is \( (6, -3) \). - The radius of the circle is \( 1 \).