#1**+3 **

Let's get the equation into the form

(x - h)^{2} + (y - k)^{2} = r^{2} where the point (h, k) is the center of the circle and r is the radius.

9x^{2} - 18x + 9y^{2} + 36y + 44 = 0

Subtract 44 from both sides of the equation

9x^{2} - 18x + 9y^{2} + 36y = -44

Divide both sides by 9

x^{2} - 2x + y^{2} + 4y = - \(\frac{44}{9}\)

Add 1 and add 4 to both sides to complete the squares on the left side

x^{2} - 2x + 1 + y^{2} + 4y + 4 = - \(\frac{44}{9}\) + 1 + 4

Factor both perfect square trinomials on the left side

(x - 1)^{2} + (y + 2)^{2} = - \(\frac{44}{9}\) + 1 + 4

Get a common denominator to combine - \(\frac{44}{9}\) + 1 + 4

(x - 1)^{2} + (y + 2)^{2} = - \(\frac{44}{9}\) + \(\frac99\) + \(\frac{36}{9}\)

(x - 1)^{2} + (y + 2)^{2} = \(\frac19\)

Now it is in the form (x - h)^{2} + (y - k)^{2} = r^{2} and we can see that...

r^{2} = \(\frac19\)

The radius is positive so take the positive sqrt of both sides

r = \(\frac13\)

hectictar May 15, 2019