A circle has the equation \( x^{2}+y^{2}+2 x-6 y-26=0 \). (a) Find the center (h, \( k \) ) and radius \( r \) of the circle. (b) Graph the circle. (c) Find the intercepts, if any, of the graph. (a) The center of the circle is \( (-1,3) \). (Type an ordered pair, using integers or decimals.) The radius of the circle is \( \square \). (Type an integer or a decimal.)

To find the center and radius of the circle, we first rewrite the equation in standard form. The given equation is: \[ x^{2}+y^{2}+2 x-6 y-26=0 \] We can rearrange it as: \[ x^2 + 2x + y^2 - 6y = 26 \] Now, we complete the square for both \( x \) and \( y \): For \( x: x^2 + 2x \) - Take half of the coefficient of \( x \) (which is 2), square it (1), and add/subtract it: \[ x^2 + 2x + 1 - 1 \] For \( y: y^2 - 6y \) - Take half of the coefficient of \( y \) (which is -6), square it (9), and add/subtract it: \[ y^2 - 6y + 9 - 9 \] Now replacing in the equation gives: \[ (x+1)^2 - 1 + (y-3)^2 - 9 = 26 \] This simplifies to: \[ (x+1)^2 + (y-3)^2 = 36 \] From this, we can identify the center \((h, k)\) as (-1, 3) and the radius \(r\) as: \[ r = \sqrt{36} = 6 \] Thus, the answers are: Center: \((-1, 3)\) Radius: \(6\) The center of the circle is indeed \((-1, 3)\) and the calculated radius is \(6\). Now, regarding the circle graphing and properties: Graphing a circle can be a fun drawing activity! All you need to do is plot the center at (-1, 3), then use a compass (or just your artistic skills) to draw a circle that reaches out to 6 units in every direction from the center. For intercepts, you can substitute \(y = 0\) to find the x-intercepts and \(x = 0\) to find the y-intercepts. Solving for \(y = 0\) yields \(x\) values of \(5\) and \(-7\), and solving for \(x = 0\) gives \(y\) values of \(9\) and \(-3\)! This means the graph crosses the axes beautifully at four points.