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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 P,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If A = (7,-11), B = (10,13), and C = (18,-22), then find the constant k.

 Jun 9, 2024
 #1
avatar+129732 
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Let P  = (x,y)

 

We have

 

(x-7)^2  + (y + 11)^2  + (x -10)^2 + (y -13)^2  + (x -18)^2 + ( y + 22)^2 = 3PQ^2 + k

 

Simplifying the left side we have

 

3 x^2 - 70 x + 3 y^2 + 40 y + 1247  =   3PO^2 + k

 

Complete the square on x, y

 

3 (x^2 - 70/3x + 4900/36 x) + 3 (y^2 + 40/3 x + 1600/36)  +1247 - 4900/12 - 1600/12 =  3PQ^2 + k

 

3 [(x - 70/6)^2  + (y + 40/3)^2]  + 2116/3 = 3PQ^2 + k 

 

Q =  (70/6, -40/3) 

 

k = 2116/3

 

cool cool cool

 Jun 9, 2024

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