# How do you find the equation of a circle that passes through (7, -1) and has a center of (-2, 4)?

May 30, 2018

${\left(x + 2\right)}^{2} + {\left(y - 4\right)}^{2} = 106$

#### Explanation:

A circle centred on the origin in the $\left(x , y\right)$ plane has equation ${x}^{2} + {y}^{2} = {R}^{2}$, where $R$ is the radius of the circle. To move any curve in $\left(x , y\right)$ to a different centre $\left({x}_{0} , {y}_{0}\right)$, simply replace $x$ by $x - {x}_{0}$ and $y$ by $y - {y}_{0}$.

So here ${\left(x + 2\right)}^{2} + {\left(y - 4\right)}^{2} = {R}^{2}$

To find the needed radius, calculate the distance between $\left(- 2 , 4\right)$ and $\left(7 , - 1\right)$. Distance is $\sqrt{{\left({x}_{a} - {x}_{b}\right)}^{2} + {\left({y}_{a} - {y}_{b}\right)}^{2}}$, i.e. $\sqrt{81 + 25} = \sqrt{106}$, so ${R}^{2} = 106$ and

${\left(x + 2\right)}^{2} + {\left(y - 4\right)}^{2} = 106$
or, multiplying out terms, which may or not be a more useful form to work with
${x}^{2} + 4 x + {y}^{2} - 8 y = 86$

May 30, 2018

${\left(x + 2\right)}^{2} + {\left(y - 4\right)}^{2} = 106$

#### Explanation:

$\text{the equation of a circle in standard form is}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(a,b)" are the coordinates of the centre and r}$
$\text{is the radius}$

$\text{the centre "=(-2,4)" and we require the radius}$

$\text{the radius is the distance from the centre to the point}$
$\text{on the circle}$

$\text{calculate r using the "color(blue)"distance formula}$

•color(white)(x)r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

$\text{let "(x_1,y_1)=(-2,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(7 , - 1\right)$

$r = \sqrt{{\left(7 + 2\right)}^{2} + {\left(- 1 - 4\right)}^{2}} = \sqrt{81 + 25} = \sqrt{106}$

$\text{substitute values into the equation}$

${\left(x - \left(- 2\right)\right)}^{2} + {\left(y - 4\right)}^{2} = {\left(\sqrt{106}\right)}^{2}$

${\left(x + 2\right)}^{2} + {\left(y - 4\right)}^{2} = 106 \leftarrow \textcolor{red}{\text{equation of circle}}$