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# 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

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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.​

Sep 6, 2023

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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

Sep 6, 2023