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# Standard equation of a circle

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archeologist found the remains of an ancient wheel,which she placed on a grid.If an arc of the wheel passes through A(-7,0),B(-3,4),and C(7,0),locate the center of the wheel,and the standard equation of the circle defining its boundary.

Jun 24, 2018

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Let  (h, k)  be the center of the circle

And the distance  from this point to each  of the points  will be  equal  (the radius)

So  we have this equation

( h - - 7)^2  + (k - 0)^2  =  ( h  - 7)^2  + (k - 0 )^2

(h + 7)^2    =  ( h - 7)^2

h^2 + 14h + 49  =  h^2 - 14h + 49

14h  =  - 14 h

28h  = 0

h  = 0

And we can solve for k, thusly

(0 - 7)^2  + ( k -0)^2  =   ( 0  - - 3)^2  + (k - 4)^2

49  + k^2  =  9  + k^2 - 8k + 16

49  = 25  - 8k

8k  =  25  - 49

8k  = - 24

k  = -3

So......the center of the circle  is   ( 0, - 3)

And the radius  is  given by  √  [ (0 - 7)^2  + ( -3 - 0)^2 ]  =  √ [ 49 + 9] = √58

So... the equation  of  the circle is

(x - h)^2  + ( y - k)^2  = r^2

( x - 0)^2  +  ( y  - - 3)^2  = 58

x^2   +  ( y + 3)^2  = 58

Here is a pic :  https://www.desmos.com/calculator/m2ugumny98

Jun 24, 2018