There is a rat race for which all the rats start from the point P (0, 3) in the XY plane. The rats have to touch x-axis and reach the point Q (15,6). The rat which reaches Q first wins. At what point should a rat touch X-axis so that the distance it travels is minimized?

This is a reflection problem in coordinate geometry. To minimize the path from P(0,3) to the x-axis and then to Q(15,6), reflect point Q across the x-axis to get Q'(15,-6). The shortest path is a straight line from P to Q', which crosses the x-axis at (5,0). This minimizes the total distance: sqrt(5² + 3²) + sqrt(10² + 6²) = sqrt(34) + sqrt(136) ≈ 18.4 units.