+219 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.
+1926 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
\(PA^2 + PB^2 + PC^2 = (x-7)^2 + (y + 11)^2 + (x-10)^2 + (y-13)^2 + (x-18)^2 + ( y + 22)^2 \)
Now, we have to simplify this gigantic equation. By expanding and combining lots of like terms, we get that
\(3 x^2 - 70 x + 3 y^2 + 40 y + 1247 \\ 3 [ x^2 - 70/3 x + y^2 + 40/3 y ] + 1247\)
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
\(3[ x^2 - 70/3 x + 4900/36 + y^2 + 40/3y + 1600/36 ] + 1247 - 4900/12 -1600/12 \\ 3 [ (x - 70/6]^2 + (y + 40/6)^2 ] + 2116 / 3\)
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! :)
+1926 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
\(PA^2 + PB^2 + PC^2 = (x-7)^2 + (y + 11)^2 + (x-10)^2 + (y-13)^2 + (x-18)^2 + ( y + 22)^2 \)
Now, we have to simplify this gigantic equation. By expanding and combining lots of like terms, we get that
\(3 x^2 - 70 x + 3 y^2 + 40 y + 1247 \\ 3 [ x^2 - 70/3 x + y^2 + 40/3 y ] + 1247\)
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
\(3[ x^2 - 70/3 x + 4900/36 + y^2 + 40/3y + 1600/36 ] + 1247 - 4900/12 -1600/12 \\ 3 [ (x - 70/6]^2 + (y + 40/6)^2 ] + 2116 / 3\)
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! :)
