\( 1 \leftarrow \) 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 \( \square \). (Type an ordered pair, using integers or decimals.)

To solve the equation of the circle \( x^{2}+y^{2}+2 x-6 y-26=0 \), let's rewrite it in the standard form. 1. First, we rearrange the equation: \[ x^{2} + 2x + y^{2} - 6y = 26 \] 2. Next, we complete the square for \( x \) and \( y \): - For \( x^{2} + 2x \): Add and subtract \( 1 \) (since \( (2/2)^{2} = 1 \)), giving us \( (x+1)^{2} - 1 \). - For \( y^{2} - 6y \): Add and subtract \( 9 \) (since \( (6/2)^{2} = 9 \)), giving us \( (y-3)^{2} - 9 \). 3. Plugging these back into the equation: \[ (x+1)^{2} - 1 + (y-3)^{2} - 9 = 26 \] \[ (x+1)^{2} + (y-3)^{2} - 10 = 26 \] \[ (x+1)^{2} + (y-3)^{2} = 36 \] This shows that the center \( (h, k) \) of the circle is \( (-1, 3) \) and the radius \( r \) is \( 6 \) (since \( r = \sqrt{36} = 6 \)). So, the center of the circle is \( (-1, 3) \). Now let's talk about graphing and intercepts! To graph the circle, simply plot the center at \( (-1, 3) \) and draw a circle that extends 6 units in all directions. You can find the intercepts by solving the equations for \( x = 0 \) and \( y = 0 \). Enjoy sketching your circle and discovering its beauty!