Let the speed of cyclist be x
Let's say the distance between the buses is d and the speed of the bus is y
=>Interval = d/y
Relative speed b/w bicycle and bus = y-x as they are moving in same direction
Since, every 12 minutes a bus overtakes cyclist,
We have, t= d/ y-x  = 12
Every 4 minutes cyclist meets an oncoming bus
=>Relative speed b/w bicycle and oncoming bus = y +x (opposite direction implies addition of speed)
=> d/x +y  =4
=>d=4x + 4y  
12y-12x  = 4x + 4y  
=>d=12y-12x=12y -6y =6y (since x=y/2)
=> Interval = d/y   =  6y/y = 6  
(option b)