In this scenario, the most appropriate option to describe the asteroid's pathway as it approaches and eventually impacts a much larger planetary body is:
The asteroid had an ellipse pathway.
As the asteroid approaches the larger body, its trajectory can be influenced by the gravity of the planet, resulting in an elliptical orbit that leads to a collision, especially if it is captured by the planet's gravity as it nears.
However, it's worth noting that if the asteroid were on a trajectory that was purely gravitationally bound and not escaping, it could also follow a parabolic or elliptical path leading to impact; in general scenarios where an object comes from far away and is drawn into a gravitational well, the pathway can often be elliptical (and in some specific cases hyperbolic if the object's speed is excess escape velocity). In your question context involving a collision, "ellipse pathway" is the most fitting choice.