# How do you write an equation for a circle passing through (-1,2), (3,4), and (2,-1)?

Jul 27, 2016

${\left(x - \frac{5}{3}\right)}^{2} + {\left(y - \frac{5}{3}\right)}^{2} - \frac{65}{9} = 0$

#### Explanation:

The circle's general equation is

$C \to {\left(x - {x}_{0}\right)}^{2} + {\left(y - {y}_{0}\right)}^{2} - {r}^{2} = 0$

This equation must be obeyed by the three points. Substituting $\left\{x , y\right\}$ for each point we have

 { ( 2 x_0 + x_0^2 - 4 y_0 + y_0^2+5 - r^2=0), ( - 6 x_0 + x_0^2 - 8 y_0 + y_0^2+25 - r^2=0), ( - 4 x_0 + x_0^2 + 2 y_0 + y_0^2+5 - r^2=0) :}

Those equations are easily handled taking the difference between the first and the second, and the first and the third, obtaining

{ (2 x_0 + y_0-5=0), (x_0 - y_0=0) :}

Solving for ${x}_{0} , {y}_{0}$ we obtain

$\left\{{x}_{0} = \frac{5}{3} , {y}_{0} = \frac{5}{3}\right\}$

$r$ is obtained putting those results in any of the equations already obtained

$r = \frac{\sqrt{65}}{3}$