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.
+34 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
