# How do you find the center (h,k) and radius r of the circle with the given equation x^2+y^2-18x+18y=-137?

Jan 20, 2016

The centre is $\left(9 , - 9\right)$ and the radius is $5$

#### Explanation:

The equation needs to be rearranged into the form ${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

First reorder the expression to group the $x$ and $y$ terms
${x}^{2} - 18 x + {y}^{2} + 18 y = - 137$

Each group then needs to be expressed as a square by completing the squares

${\left(x - 9\right)}^{2} - {9}^{2} + {\left(y + 9\right)}^{2} - {9}^{2} = - 137$
${\left(x - 9\right)}^{2} + {\left(y + 9\right)}^{2} = - 137 + 81 + 81 = 25$

The circle is then ${\left(x - 9\right)}^{2} + {\left(y + 9\right)}^{2} = 25$

The centre is therefore $\left(9 , - 9\right)$ and the radius is $5$