OE is the center line on the hypotenuse of right triangle AOB,

? So OE= 1/2 AB.

Therefore, the OE length is fixed at 1.

DE is the hypotenuse of a right triangle DAE,

So the length of DE is also a fixed value √2.

In the process of exercise, there is always OE+DE & gt；; OD (the sum of two sides of triangle ODE is greater than the third side);

But there is only one special case, and that is

When point E coincides with the intersection of OD and AB, that is, points O, E and D are three points and one line,

OE+DE=OD

At this time, OD has a maximum value =√2+ 1.

Therefore, choose a.