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# coordinates

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What are the coordinates of the center of the circle passing through the points (1, 2), (3,-4), and (5,-6)?

Apr 12, 2020

#1
+23623
+1

The distance from each of the point to the center has to be the same (radius)

pick two points and calculate the distances and equate them

1,2    and   3,-4

(x-1)^2 +  (y-2)^2     =    (x-3)^2 + (y+4)^2

x^2 - 2x + 1     + y^2 -4y + 4   =    x^2 -6x+9      + y^2+8y + 16

4x                              - 12y   -20 = 0

now pick   3 , -4      and   5, -6

(x-3)^2  + (y+4)^2    =   (x-5)^2  + (y+6)^2

x^2 -6x + 9    +    y^2 + 8y + 16       =   x^2 -10x + 25    +    y^2 + 12y +36

4x                              -4y     -36  = 0          two red equations ...solve for x and y       multiply THIS one by -1 and add to the one above to get

-8y + 16 = 0       y = 2

then x = 11              center is   11,2

Apr 12, 2020
#2
+23623
0

Equation of circle will be     (x-11)^2  + (y-2)^2 = 10^2

Here is graph:

https://www.desmos.com/calculator/oqi21ttyz8

Apr 12, 2020