Let A(3,-1) and B(1,1) be the end points of line segment AB. Let P be the middle point of the line segment AB. Let Q be the point situated at a distance \sqrt{2} units from P on the perpendicular bisector line of AB. What are the <strong>POSSIBLE</strong> coordinates of Q?

Explanation

Midpoint P = (2,0). The slope of AB is \frac{1-(-1)}{1-3} = -1. The perpendicular bisector has slope 1 and passes through (2,0), giving the line y = x-2. Any point on this line is (x, x-2). The distance PQ = \sqrt{2} \implies (x-2)^2 + (x-2-0)^2 = 2 \implies 2(x-2)^2 = 2 \implies x-2 = \pm 1 \implies x = 3 or 1. The points are (3,1) and (1,-1). Only (3,1) is in the options.

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