Bear in mind that:
1) The farther away an Earth-threatening object is, the easier it is to prevent from hitting. A small change of angle at a great distance will add up to a very wide miss. At closer range, much greater change of direction is required to miss.
2) It is much easier to divert an object in motion than to stop it, and if it is large enough to do serious damage, also easier than destroying it. That usually means you want to bump it from the side nearer Earth, so that it moves away.
3) You must avoid hitting the Earth and the Moon. (See Thundarr the Barbarian.) Sadly that makes the target much bigger, because the object has to go outside the territory occupied by what is basically a binary planet system, necessitating a bigger change of direction. And don't let it near the LaGrange points either (including the Earth/Moon and Earth/Sun sets). That makes early identification and interception essential. Forget Hollywood's last-minute salvation attempts. This is a problem to solve with math and copious explosives.