Personally....knowing the human nature of the trash that has his arm around my loved one's neck and a knife to their throat....I would take the shot to the face.
I am going to guess that there is a 95% chance the BG will kill the hostage if I do nothing. I will take my chances that a good shot to the forehead will change his way of thinking and in my mind there is less of a chance that my loved one will be hurt by some muscle twitch than by his free will.
I think that reasoning with the guy in a calm manner and saying "if you put down the knife and don't harm ____ I will let you leave...I don't want to shoot you or anyone else". However, you aren't leaving with ______". I think that MOST bad guys will take the easy way out and drop the knife. After they did, I would shoot him several times for spite.