The equation of a circle is \( x^{2}+y^{2}-6 x+4 y=k \). (1) Determine the coordinates of the centre of the circle. (2) If the radius of the circle is 13 units, determine the value of \( k \).

To analyze the equation of the circle given by \[ x^{2} + y^{2} - 6x + 4y = k, \] we can rewrite it in standard form by completing the square for both \(x\) and \(y\). ### Step 1: Completing the square **For \(x\):** The terms involving \(x\) are \(x^{2} - 6x\). To complete the square: \[ x^{2} - 6x = (x - 3)^{2} - 9. \] **For \(y\):** The terms involving \(y\) are \(y^{2} + 4y\). To complete the square: \[ y^{2} + 4y = (y + 2)^{2} - 4. \] ### Step 2: Rewrite the equation Substituting these completed squares back into the equation gives: \[ (x - 3)^{2} - 9 + (y + 2)^{2} - 4 = k. \] Combining the constants: \[ (x - 3)^{2} + (y + 2)^{2} - 13 = k. \] Rearranging this, we have: \[ (x - 3)^{2} + (y + 2)^{2} = k + 13. \] ### Step 3: Identify the center From the standard form of the circle \((x - h)^{2} + (y - k)^{2} = r^{2}\), we can identify the center \((h, k)\) of the circle. Here, the center is: \[ (3, -2). \] ### Step 4: Determine the value of \(k\) when the radius is 13 The radius \(r\) of the circle is given by the equation: \[ r^{2} = k + 13. \] Given that the radius is 13, we have: \[ r^{2} = 13^{2} = 169. \] Setting this equal to the expression we derived: \[ 169 = k + 13. \] ### Step 5: Solve for \(k\) To find \(k\), we rearrange the equation: \[ k = 169 - 13 = 156. \] ### Summary of Results 1. The coordinates of the center of the circle are \((3, -2)\). 2. The value of \(k\) when the radius is 13 units is \(156\).