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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 = , B = , and C = , then find the constant k.

 Mar 16, 2023
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The coordinate plane contains points A, B, and C. The constant k then changes to 32.

 

Let P be (x,y)

= Pa^2+Pb^2+pc^2

= (x-4)^2+(y+1)^2+(x-6)^2+(y-2)^2+(x+1)^2+(y-2)^2

= 3x^2+3y^2-18x-6y+64

= 3(x^2+y^2-6x-2y)+62

= 3((x-3)^2+(y-1)^2)+32

This shows that if Q = (3,1), Pq^2=(x-3)^2+(y-1)^2

And so pa^2+pb^2+pc^2=3pq^2+32

k = 32

 Mar 17, 2023

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