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Points A, B, and C are given in the coordinate plane. There exists a point Q and a constant k such that for any point PPA2+PB2+PC2=3PQ2+k
If A=(2,5), B=(3,4), and C=(1,2), then find the constant k.

 Jan 26, 2024
 #1
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Let P=   (x,y)

 

PA^2  + PB^2 + PC^2  =

 

(x -2)^2 + (y + 5)^2  + (x -3)^2  + (y  -4)^2  + (x -1)^2 + (y + 2)^2  = 

 

3x^2 -12x + 3y^2 + 6y + 59  = 

 

3 ( x^2 - 4x + y^2  + 2y) + 59      complete the square on x ,y

 

3 [  x^2 - 4x + 4  + y^2 +2y + 1]  +  59  - 12 - 3

 

3 [ (x - 2)^2  + ( y + 1)^2 ] + 44

 

Q = (2, -1)

 

k =  44

 

 

cool cool cool

 Jan 26, 2024
edited by CPhill  Apr 7, 2024

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