# How do you write the standard from of the equation of the circle given center is at (-1,2) and the point (3,4) lies on the circle?

Mar 9, 2018

${\left(x + 1\right)}^{2} + {\left(y - 2\right)}^{2} = 20$

#### Explanation:

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

•color(white)(x)(x-a)^2+(y-b)^2=r^2

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

$\text{here } \left(a , b\right) = \left(- 1 , 2\right)$

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

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

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

$\text{let "(x_1,y_1)=(-1,2)" and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 4\right)$

$\Rightarrow r = \sqrt{{\left(3 + 1\right)}^{2} + {\left(4 - 2\right)}^{2}} = \sqrt{16 + 4} = \sqrt{20}$

$\Rightarrow {\left(x - \left(- 1\right)\right)}^{2} + {\left(y - 2\right)}^{2} = {\left(\sqrt{20}\right)}^{2}$

$\Rightarrow {\left(x + 1\right)}^{2} + {\left(y - 2\right)}^{2} = 20 \leftarrow \textcolor{red}{\text{equation of circle}}$