Load line and voltage or current gain
I think, you are using the diagram not in the right manner.


Steeper load line: When the base current varies between 30µA and 50µA, the collector current change is between 2.5....and 5mA. This gives a current gain of 2.5E3/20=125. 
The corresponding Vce change is 14-9=5Volt.
Lower load line: For the same Ib variation, we have, of course, the same variation in collector current and  hence, the same current gain. However, the Vce variation now is between app. 0.5V and 9V (9-0.5=8.5V). 


I think, that`s all you can derive from this graph. So, I do not understand the question which assumes Vce=const.

Comment/update: You are asking "Why does a steeper load line gives higher current gain...."?

More or less by accident, I have found a formula which shows how much the current gain goes down for a smaller slope of the load line (when the sum (Rc+Re) increases.):

ic/ib=hfe/[1+(Rc+Re)/ro] with hfe=beta (short circuit current gain)  and ro=d(Vce)/d(Ic)