Option 1: According to the given equations x = 1 sin(ωt) and y = b cos(ωt), there is no mention or indication of the radius `a`. Therefore, the option 1 of a circular path with radius `a` is incorrect.

Option 2: Similar to option 1, the given equations do not mention or indicate a radius `b`. Hence, option 2 of a circular path with radius `b` is also incorrect.

Option 3: The equations x = 1 sin(ωt) and y = b cos(ωt) form the equations of an ellipse. By substituting different values of x and y in these equations, an elliptical path can be observed. Therefore, option 3 of an elliptical path is the correct answer.

Option 4: The equations x = 1 sin(ωt) and y = b cos(ωt) do not represent a straight line. As per these equations, the particle`s motion follows a periodic oscillation along both the x and y-directions. Thus, the option 4 of a straight line is incorrect.

In conclusion, the correct option is 3, which represents an elliptical path for the motion of the particle.