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

michaelcai  Oct 30, 2017

Best Answer 

 #1
avatar+7155 
+2

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.

hectictar  Oct 31, 2017
 #1
avatar+7155 
+2
Best Answer

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.

hectictar  Oct 31, 2017

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