Determine the coordinates of the point \(P\) on the line \(y=-x+6\) such that \(P\) is equidistant from the points \(A(10,-10)\) and \(O(0,0)\) (that is, so that \(PA=PO\)). Express your answer as an ordered pair \((a,b)\).
+130536 May be several ways to do this
Notice that the slope of a segment drawn between O and A = [ -10- 0] /[10-0] = -10 /10 = -1
Which is the same slope of the line y = -x + 6
We can find the midpoint of OA and write an equation of the line through this midpoint that is perpendicular to y = - x + 6
The intersection of these two lines will give us the "P" we are looking for
Midpoint of OA = (5, - 5)
Equation of a line through this point perpendicular to y = -x + 6 is
y = (x -5) - 5
y = x - 10
Find the x coordinate of the intersection of these lines
-x + 6 = x - 10
2x = 16
x = 8
And y = -8 + 6 = -2
So P = (8, -2)
And we can see that this point makes PO = PA
