# In the xy-plane, the graph of 2x^2-6x+2y^2+2y=45 is a circle? What is the radius of the circle?

Mar 24, 2017

$\text{radius } = 5$

#### Explanation:

The standard form of the $\textcolor{b l u e}{\text{equation of a circle}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where (a ,b) are the coordinates of the centre and r, the radius.

The first step is to divide all terms by 2

$\Rightarrow {x}^{2} - 3 x + {y}^{2} + y = \frac{45}{2}$

Using the method of $\textcolor{b l u e}{\text{completing the square}}$

${x}^{2} - 3 x \textcolor{red}{+ \frac{9}{4}} + {y}^{2} + y \textcolor{red}{+ \frac{1}{4}} = \frac{45}{2} \textcolor{red}{+ \frac{9}{4} + \frac{1}{4}}$

$\Rightarrow {\left(x - \frac{3}{2}\right)}^{2} + {\left(y + \frac{1}{2}\right)}^{2} = 25 \leftarrow \textcolor{b l u e}{\text{ standard form}}$

${r}^{2} = 25 \Rightarrow r = 5$