# How do you determine the standard form of the equation of a circle with center at (4,-1) and passing through (0, 2)?

Jun 2, 2016

${\left(x - 4\right)}^{2} + {\left(y - 1\right)}^{2} = 25$

#### Explanation:

The standard form of the equation of a circle is.

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

To establish the equation , we require the radius. The length of the line joining the centre and point on the circumference (0 ,2) is the radius.
Using the $\textcolor{b l u e}{\text{distance formula}}$ on the points (4 ,-1) and (0 ,2)

$d = \sqrt{{\left(0 - 4\right)}^{2} + {\left(2 + 1\right)}^{2}} = \sqrt{25} = 5 \text{ equals r}$

$\Rightarrow {\left(x - 4\right)}^{2} + {\left(y - 1\right)}^{2} = 25 \text{ is the equation of the circle }$