# How do you find the center and radius of the circle given x^2+y^2-18x-18y+53=0?

Oct 13, 2016

Complete the square for the x and y terms to find the center $= \left(9 , 9\right)$ and the radius $r = \sqrt{109}$.

#### Explanation:

${x}^{2} + {y}^{2} - 18 x - 18 y + 53 = 0$

Use a method called "Completing the Square".

1) Group the x terms and y terms. Move the constant term to the right side of the equation by subtracting 53 from both sides.

$\left({x}^{2} \textcolor{\lim e g r e e n}{- 18} x \textcolor{w h i t e}{a a a}\right) + \left({y}^{2} \textcolor{m a \ge n t a}{- 18} y \textcolor{w h i t e}{a a a}\right) = - 53$

2) Divide the coefficient of the x term by 2 and then square it.

$\frac{\textcolor{\lim e g r e e n}{- 18}}{2} = - 9 \textcolor{w h i t e}{a a a} {\left(- 9\right)}^{2} = \textcolor{red}{81}$

3) Add the result to both sides.

$\left({x}^{2} \textcolor{\lim e g r e e n}{- 18} x + \textcolor{red}{81}\right) + \left({y}^{2} \textcolor{m a \ge n t a}{- 18} y \textcolor{w h i t e}{a a a}\right) = - 53 + \textcolor{red}{81}$

4) Divide the coefficient of the y term by 2 and then square it.

$\frac{\textcolor{m a \ge n t a}{- 18}}{2} = - 9 \textcolor{w h i t e}{a a a} {\left(- 9\right)}^{2} = \textcolor{b l u e}{81}$

5) Add the result to both sides.

$\left({x}^{2} - 18 x + \textcolor{red}{81}\right) + \left({y}^{2} - 18 y + \textcolor{b l u e}{81}\right) = - 53 + \textcolor{red}{81} + \textcolor{b l u e}{81}$

Factor each set of parentheses. Note that the $- 9$ in factored form is the same number you got when dividing the coefficient of the middle term.

$\left(x - 9\right) \left(x - 9\right) + \left(y - 9\right) \left(y - 9\right) = 109$

Rewrite as the square of a binomial.

${\left(x - 9\right)}^{2} + {\left(y - 9\right)}^{2} = 109$

Compare this equation to the equation of a circle
${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$
where $\left(h , k\right)$ is the center and $r$ is the radius.

In this example, the center $\left(h , k\right) = \left(9 , 9\right)$ and the radius $r = \sqrt{109}$