If two persons A and B start at the same time in opposite directions from two points and complete the journeys in a and b hours respectively after passing each other,then A' speed /B's speed= √b/√a. Can I get the proof for this.

Dear Student,
Assume the speed of A is Sa and that of B is Sb. Consider the distance between the two is x and let y be the distance from B to their meeting point. Now make two equations based on following conclusion,
1. time taken by both A and B to reach the meeting point will be equal.
(x-y)/Sa = y/Sb --- (1)
2. given that a takes a hours to reach B after meeting (a = y/Sa). Similarly b = (x-y)/Sb.
Now replace the value of x-y and y in (1) by as given in (2). You can easily solve now. Thanks.