If we set a and b different but both positive, a=5 and b=2 and hold k constant at 1 and set t from -2 to 2 we get a graph that appears as
Some coordinates of the line are: (3,0) , (4,1) , ... ,(7,4). This would appear to also be a slope of one.
This still should be what we expected to see if we are assuming that k is associated with the slope. So even changing from a and b equal, the coordinates more, but the slope does not appear to be affected.
Let's try one more change of k say k=-2 we get:
Some coodinates of the line are now: (3,6) , (4,4) , ... , (7,-2). This would suggest a slope of -2.
This furthers our point even more that k affects the slope of the line.