A piece of broken plate was dug up in an archaeological site. It was put on top of a grid with the arc of a plate passing through A(-7,0), B(1,4) and C(7,2). Find its center and the standard equation of the circle describing the boundary of the plate.
+128485 We have the following equations where (x,y) is the center of the circle and r^2 = the radius^2
(x + 7)^2 + y^2 = r^2
(x - 1)^2 + (y - 4)^2 = r^2
(x - 7)^2 + (y - 2)^2 = r^2
Simplify these
x^2 + 14x + 49 + y^2 = r^2 (1)
x^2 - 2x + 1 + y^2 - 8y + 16 = r^2 (2)
x^2 - 14x + 49 + y^2 - 4y + 4 = r^2 (3)
Subtract (1) from (2)
-16x - 48 -8y + 16 = 0
-16x -8y - 32 = 0 divide through by -8
2x + y + 4 = 0
2x + y = -4 → y = -2x - 4 (4)
Subtract ( 3) from (2)
12x - 48 - 4y + 12 = 0 divide through by 4
3x - 12 - y + 3 = 0
y = 3x - 9 (5)
Set (4) = (5)
-2x - 4 = 3x -9
5 = 5x
x = 1
y = 3(1) - 9 = -6
The center is ( 1, -6)
Using (1)
(1 + 7)^2 + (-6)^2 = r^2
8^2 + 36 = r^2
100 = r^2
The equation for the circle is
(x - 1)^2 + ( y + 6)^2 = 100
x^2 - 2x + 1 + y^2 + 12y + 36 = 100
x^2 + y^2 - 2x + 12y - 63 = 0
