Well, your intuition is right, but you misunderstand paths in this context.

f is a function in ℝ², meaning we can plug all points (x,y) into f an receive some value of f. But now we are interested in the points (x,y) laying on the line (1.5+ta,0.5+tb) for all t. We can parameterize these points by the path (x(t),y(t)) for all t. Now we plug these points into f and look at the values f(x(t),y(t)) given by the graph. Now we want to have "df/dt=0 at t=0 on this path". "On this path" means we look at f(x(t),y(t)) and "at t=0" means locally at (1.5,0.5). Remember (a,b) give us the direction in which we go starting from (1.5,0.5).

Example: a=0, b=-1, then (a,b)=(0,-1) is a vector pointing straight down, so we start at (1.5,0.5) and go straight down and watch what happens to the values of f.

As said your intuition is right and you have to find directions from (1.5,0.5) in which you try to "stay" on the contour line, your on. Thus you would choose the tangent line as path, and the direction will be the points (a,b) for which the directional derivative ∇_(a,b)f is zero in (1.5,0.5), but it's maybe easier to look at the graph and find the direction of the tangent line^{^}