+2651 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.
+129829 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
