How do you write the equation of the circle with center (2,4) and containing the point (-2,1)?

Apr 27, 2016

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

Explanation:

The equation of a circle with center $\left(h , k\right)$ and radius $r$ is:
${r}^{2} = {\left(x - h\right)}^{2} + {\left(y - k\right)}^{2}$

We are told that the center of this circle is $\left(2 , 4\right)$, so
${r}^{2} = {\left(x - 2\right)}^{2} + {\left(y - 4\right)}^{2}$

However, we don't know the radius of this circle. Luckily, we are also told that this circle contains the point $\left(- 2 , 1\right)$, so we will use that information to find $r$:
${r}^{2} = {\left(x - 2\right)}^{2} + {\left(y - 4\right)}^{2}$
${r}^{2} = {\left(- 2 - 2\right)}^{2} + {\left(1 - 4\right)}^{2}$
${r}^{2} = 16 + 9$
${r}^{2} = 25$
$r = 5$

The radius of the circle is $5$, which means the equation of the circle is:
$25 = {\left(x - 2\right)}^{2} + {\left(y - 4\right)}^{2}$