Let a and b be distinct complex numbers on the unit circle. Find the maximum possible value of .

idk,

but if  a=1 and b=-1 than it would be     2/2 = 1

I suspect that is the biggest but I do not really know.  I am just guessing.

Since they are both on the unit circle, a and b differ only in angle.  Without loss of generality we can set the angle of b to be zero, so that b = 1.

Then $|\frac{a-b}{1-\bar{a} b}|=|\frac{a-1}{1-\bar{a}}|$  

If we let  $a=e^{i\theta}$  then $|\frac{a-1}{1-\bar{a}}|=|\frac{e^{i\theta}-1}{1-e^{-i\theta}}|=|e^{i\theta}|=1$

So, as long as $\theta \ne 0$  the ratio is 1