Not homework. I'm studying from a circuits book on my own.


Why is there no current through the branch with the 9 Ohm resistor?

Mod: lightened you image.E

 

All current Flow Paths follow a closed loop. There is no loop for a current flow through the 9 Ohm resistor.
If you short the terminals a and b, a closed loop is created, there maybe a current flow that needs to be worked out.

 

I dont know if this enhanced pic helps too much but here it is. In the future please try to post pictures that are clear which may mean taking a second picture of the book page with better lighting.

As to the problem itself, the previous post outlined it well. You need a closed path in order for current to flow.

Interestingly though, if you open circuit the 10 ohm and 5 ohm resistors then you have a real conundrum because then you have a current source inside an open circuit. You may want to leave this question for a later date though.