+101 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 \(A = (2,-5),\) \(B = (3,4),\) and \(C = (1,-2)\), then find the constant \(k\).
+129850 Let P= (x,y)
PA^2 + PB^2 + PC^2 =
(x -2)^2 + (y + 5)^2 + (x -3)^2 + (y -4)^2 + (x -1)^2 + (y + 2)^2 =
3x^2 -12x + 3y^2 + 6y + 59 =
3 ( x^2 - 4x + y^2 + 2y) + 59 complete the square on x ,y
3 [ x^2 - 4x + 4 + y^2 +2y + 1] + 59 - 12 - 3
3 [ (x - 2)^2 + ( y + 1)^2 ] + 44
Q = (2, -1)
k = 44
