Here are some paladin archetypes. As for their responses in a hostage situation, I'd need information about the circumstances of the specific situation before being able to suggest ways they might react.
The Courtier:
The courtier is the gentleman paladin, living at the court rather than in the saddle. He is likely to be as skilled at diplomacy, dancing and writing poetry as wielding his sword, and may emphasise personal grooming, courtly etiquette and fine living as much as any other qualities. Often serving as the political advisor to a particular ruler, he strives to make his liege achieve a simultaneously moral and effective reign, where the populace is kept happy while being led in the right directions. In fact, he may believe that he can lead the battle against evil more effectively through words than through actions. Of course, when battle is necessary, he is as ready as any other paladin, although he will usually fight with an emphasis on finesse and style. The courtier is almost never an adventuring paladin, leaving the court only due to the most extreme reasons.
The Crusader:
The crusader is a paladin who faces the world with his sword. He is usually at his best on the battlefield, leading the war of good against evil in the most direct way possible. The crusader fights evil without asking quarter and rarely gives any. Focussed and driven, he is often a loner (although he may also serve as part of a like-minded organisation), fighting a holy war against evil (in general), although he usually works against a specific group/individual/species, etc. Crusaders are commonly adventuring paladins, often having some tragic incident in their past which drives them onto their path, in which case a strong desire for revenge drives them. Ironically, they sometimes stumble off the paladin’s path by enacting the very evils they seek to destroy, in a too reckless approach to their aims.
The Expatriate
The expatriate is always a solitary paladin, who has lost his patronage or parent organisation, either due to its dissolution or because he has forsaken it or been dismissed. This loss defines him, is the defining aspect of his character, although it is never his fault, since he would not then retain his divinely-granted abilities. Having lost a sense of belonging, the expatriate is often bitter and cynical. He remains devoted to his beliefs, perhaps especially so since he sees himself as the last one upholding them, but sees his own quest as ultimately in vain. The battle of good against evil appears as hopeless as it is worthwhile, and the world around him appears permeated by the negative aspects of existence. Of all paladins, the expatriate is perhaps most aware of his role as a seeker for perfection in a flawed world. He is commonly an adventuring paladin.
The Healer
The healer is a paladin who uses his divinely-granted abilities to aid the poor and the diseased. He often eschews the more martial aspects of the paladin’s role, taking up arms only to protect his charges. Often working in conjunction with clerics of his (or an allied) faith, his primary goal is to care for others. Along with healing, he often provides both spiritual and material aid to those he meets. Self-aggrandisement and glory are even further from his thoughts than from those of the majority of other paladins. While the healer is often found far afield, he is rarely an adventurer. Usually he seeks out a rural community to aid with his healing abilities, before moving on to another which is more needful of his aid, and so on. In rare cases, he may become a permanent part of a small community which is closer to his heart than the grandest of royal courts.
The Inquisitor
The inquisitor is perhaps the paladin most feared by all evildoers, and strangely, by many who are good as well. Usually working as part of a larger organisation, he relentlessly seeks out and destroys all that he sees as evil. For the inquisitor, there is no acceptable path except the straight and the narrow, and all who stumble off it are deserving of chastisement. Since much evil is concealed or existing under false pretences, the inquisitor’s main role is to reveal it, and then destroy it. For the inquisitor, the end (the greater good) is always more important the means, although he strives to keep the latter lawful. This emphasis leads to the fact that the inquisitor, like the crusader, runs the risk of losing his powers through too rigorous and unthinking an approach.
The Knight-Errant
The knight-errant is the quintessential adventuring paladin, travelling out into the world in order to find, battle and defeat evil. Often young, naïve and optimistic, the knight-errant does not seek out a specific foe, but wishes to combat evil in general. While possibly owing loyalty to a specific faith or ruler, he often works alone, sometimes in the company of a group of adventurers. Usually outgoing and amiable, his interest in the surrounding world knows no limits. As part of his apparent naïveté, he is constantly cheerful, confident of the correctness of his beliefs, and sees each day as another opportunity to further the cause of good. Reliable in combat, and a boon companion out of it, the knight-errant is often the perfect member of an adventuring party.
The Militarist
The militarist paladin is a man of war, usually heading the military branch of a particular faith or serving in a high post in an army. His efforts are completely aimed at defeating evil on the battlefield, and all of his training is dedicated to that end. Inspirational and charismatic, he is a splendid leader of men, inspiring them to impossible feats by personal example. As a warrior par excellence, he is a symbol of the martial ability his followers must seek to achieve. Due to his paladin ethos, he ensures that his followers act and fight in accordance with the proper rules of warfare. While this emphasis on propriety and honour on the battlefield sometimes leads to disagreements with the more pragmatic of his superiors, the militarist’s undoubted ability usually carries the day, just as it does on the field of battle.
The Votary
The votary is the purely religious paladin, who has dedicated his life and efforts to his faith. Whether a member of an established church (where his allegiance is always to his deity before the institution), or a lonely anchorite in a secluded cell in the wilderness, the votary concentrates body and soul on understanding his god’s wishes and enacting his commands. Although he is as martially capable as any other paladin, he will usually fight only to protect his faith, rather than to aggressively carry it forth. Philosophical and introspective, the votary’s life is spent in fruitful work, usually within the bounds of the establishment he calls home. Like the courtier, the votary is almost never an adventuring paladin, venturing forth from his home only for the most important of reasons.