Determine the coordinates of the point $P$ on the line $y=-x+6$ such that $P$ is equidistant from the points $A(10,-12)$ and $O(2,8)$ (that is, so that $PA=PO$). Express your answer as an ordered pair $(a,b)$.
Let P=(x,−x+6). We have
Note that . So the slope of perpendicular bisector of AO is .
Midpoint of AO is . Since PA = PO, P lies on the perpendicular bisector of AO. Then,
The point is .
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