The equation of a line through the point 1,2 whose distance from the point 3,1 has the greatest value is

Obviously, the line through Q is at the greatest distance from point P when it is perpendicular to PQ. Now, the slope of line PQ. Now, the slope of line PQ is mPQ=−1/2. Then the slope of perpendicular line passing through Q is y−2=2x−1or 2x−y=0