Find the center of the circle passing through the points $(-1,0)$, $(1,0)$, and $(3,1)$. Express your answer in the form '$(a,b)$.'

The distance from the center of the circle to each of the  3  points will be the same.

We can call this distance  " r "  for radius.  So we know...

 

(-1 - a)² + (0 - b)²  =  r²

(1 - a)² + (0 - b)²  =  r²            And...

(3 - a)² + (1 - b)²  =  r²

 

Set the first two values of  r²  equal to each other.

 

(-1 - a)²  + b²  =  (1 - a)² + b²

                                                      Subtract  b²  from both sides.

(-1 - a)²  =  (1 - a)²

                                                      Take the  ±  square root of both sides.

-1 - a  =  ±(1 - a)

 

-1 - a  =  1 - a     or     -1 - a  =  -1 + a     →     a  =  0

 

Using this value for  a , set the second and third values of  r²  equal to each other.

 

(1 - 0)² + b²  =  3² + (1 - b)²           Solve this for  b .

 

1 + b²  =  9 + 1 - 2b + b²

 

1  =  10 - 2b

-9  =  -2b

9/2  =  b

 

So the center of the circle is  (0, 9/2) .  Here's a graph.