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.
+128631 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
