Christmas Contest - Team Round - Problem 20

On the xy plane, a robot can move along vertical or horizontal directions to the destination, or it can move in a straight line directly to the destination. It cannot do both. If the robot moves along vertical and horizontal lines only, it travels at 1.4 m/s.

If it moves in any straight line, it travels at 1 m/s. The region of points in the first quadrant for which a robot at the origin that moves in any straight line will get to the destination faster than the one that moves only vertically or horizontally is bounded by two lines.

Let the angle between the lines be \theta. Let \tan{\theta} be \dfrac{a}{b} where \gcd(a,b)=1. Find a+b.