+216 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.
+1953 Let's set up some variables to solve this problem.
Let's first set that P=(x,y)
Now, from the problem, we can write the complicated equation of
PA2+PB2+PC2=(x−7)2+(y+11)2+(x−10)2+(y−13)2+(x−18)2+(y+22)2
Now, we have to simplify this gigantic equation. By expanding and combining lots of like terms, we get that
3x2−70x+3y2+40y+12473[x2−70/3x+y2+40/3y]+1247
Now, we simplfy complete the square for x and y to achieve our x and y values for point Q, which will be crucial.
Completing the square, we have
3[x2−70/3x+4900/36+y2+40/3y+1600/36]+1247−4900/12−1600/123[(x−70/6]2+(y+40/6)2]+2116/3
Now, from this, we can tell what Q has for their x and y value.
We have that Q = (70/6, -20/3)
Thus, we know that
k=2116/3
So 2116/3 is our final answer.
Thanks! :)
+1953 Let's set up some variables to solve this problem.
Let's first set that P=(x,y)
Now, from the problem, we can write the complicated equation of
PA2+PB2+PC2=(x−7)2+(y+11)2+(x−10)2+(y−13)2+(x−18)2+(y+22)2
Now, we have to simplify this gigantic equation. By expanding and combining lots of like terms, we get that
3x2−70x+3y2+40y+12473[x2−70/3x+y2+40/3y]+1247
Now, we simplfy complete the square for x and y to achieve our x and y values for point Q, which will be crucial.
Completing the square, we have
3[x2−70/3x+4900/36+y2+40/3y+1600/36]+1247−4900/12−1600/123[(x−70/6]2+(y+40/6)2]+2116/3
Now, from this, we can tell what Q has for their x and y value.
We have that Q = (70/6, -20/3)
Thus, we know that
k=2116/3
So 2116/3 is our final answer.
Thanks! :)
