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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.

 May 29, 2024
 #1
avatar+129487 
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Let P = (x,y)

 

PA^2 + PB^2 + PC^2   =

 

(x-7)^2 + (y + 11)^2  + (x-10)^2 + (y-13)^2 + (x-18)^2 + ( y + 22)^2    .....simplifying, we have

 

3 x^2 - 70 x + 3 y^2 + 40 y + 1247

 

3 [ x^2 - 70/3 x + y^2 + 40/3 y ] + 1247       complete the square on  x,y

 

3[ x^2 - 70/3 x + 4900/36 + y^2 + 40/3y + 1600/36 ] + 1247 - 4900/12 -1600/12

 

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

 

Q = (70/6, -40/6 ) =  (35/3 , - 20/3)

 

k = 2116 / 3

 

cool cool cool

 May 30, 2024

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