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# Help! and include explanation

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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 = (4, -4), B = (3, 5), and C = (-1, 2), then find the constant k.

Apr 7, 2024

#1
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Here's a similar problem

https://web2.0calc.com/questions/pls-help-asap_78#r1

See if you can take it  from  here

Apr 7, 2024
edited by CPhill  Apr 7, 2024
#2
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Sorry for the trouble but i cant really solve the problem with the other problem (if that makes sense) , can you help?

rerebas  Apr 7, 2024
#3
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Let P = (x,y)

PA^2  + PB^2 + PC^2   =

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

Simplifying this we  get

3 x^2 - 12 x + 3 y^2 - 6 y + 71

3 [ x^2 - 4x + y^2 - 2y ] + 71        complete the square on x, y

3 [ x^2 -4x + 4 + y^2 -2y + 1 ]  + 71 - 12 - 3

3 [ ( x - 2)^2  + (y - 1]^2 ]  +  56

Q = (2, 1)

k  =  56

CORRECTED

Apr 7, 2024
edited by CPhill  Apr 7, 2024
edited by CPhill  Apr 7, 2024
#4
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Apparently, thats wrong, all the answers i turned in are :50, 71 26/3 64 62

Apr 7, 2024
#5
+129829
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