IMPORTANT CONCEPT: If 3 points are collinear, those points lie on the same line.
So, the slope between any two points on the line will be equal to any other two points on the line.

nguyendinhtuong has already demonstrated this in the above post, by equating the slope between (a,0) and (1,1) with the slope between (0,b) and (1,1)
I just want to follow up on this and show that the strategy works with any 2 pairs of points.

Let's find the slope between (a,0) and (0,b) and the slope between (0,b) and (1,1)

Slope between (a,0) and (0,b) = (b-0)/(0-a) = b/(-a)
Slope between (0,b) and (1,1) = (1-b)/(1-0) = (1-b)/1

So, it must be the case that b/(-a) = (1-b)/1
Cross multiply to get: (1)(b) = (-a)(1-b)
Simplify: b = -a + ab
Add a to both sides: b + a = ab
Divide both sides by ab to get: b/ab + a/ab = ab/ab
Simplify: 1/a + 1/b = 1

Answer: C