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Let A(-8,4) and B(6,6).  Point C(0,k) is such that the circumcircle of triangle ABC has equation x^2 + y^2 + 2x - 10y - 24 = 0.  Find all possible values of k.

 Nov 18, 2019
 #1
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We can find the  center and radius of the circle by completing the square on x and  y

 

x^2 + 2x + 1  + y^2 -10y + 25  = 24 + 1 + 25

(x + 1)^2  + (y - 5)^2  = 50

 

The center is  ( -1, 5)  and the radius is sqrt (50)

 

And we can find the possible values for k by solving this :

 

(-1 -0)^2 + ( 5 - k)^2 = 50    

 

1  + k^2 - 10k + 25  = 50

 

k^2 - 10k + 26  =  50

 

k^2  - 10k - 24    = 0     factor

 

(k - 12) (k + 2)  = 0

 

Set both factors to 0 and solve for k and we get that

 

k = 12     or   k   = -2

 

See the image here :  

 

 

 

cool cool cool

 Nov 18, 2019

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