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This situation just came up in a game (caster is the halfling with the ponytail). I can't tell if the Large enemy halfway behind a door is affected by burning hands that starts out pointing directly at that closed half of the door.
A burst spell affects whatever it catches in its area, including creatures that you can't see. It can't affect creatures with total cover from its point of origin (in other words, its effects don't extend around corners). The default shape for a burst effect is a sphere, but some burst spells are specifically described as cone-shaped. A burst's area defines how far from the point of origin the spell's effect extends.
The enemy doesn't have total cover (italicized), but the spell does have to go around a corner (bolded) the way I see it.
It's a given that the enemy gets normal cover (+2 Reflex) from the closed part of the door, but does the spell affect it at all?

Gallant Armor |
If you don’t have line of effect to your target (that is, you cannot draw any line from your square to your target’s square without crossing a solid barrier), he is considered to have total cover from you. You can’t make an attack against a target that has total cover.
For a cone with a starting edge, either corner of the starting edge counts as a point of origin for the purposes of determining total cover. If either corner has line of effect then the enemy doesn't have total cover. In your example the enemy would not have total cover as the northern corner has line of effect to the enemy.

blahpers |
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An irksome thing: The "big creatures and cover" rules specify that you can select any of the big target's square to decide cover vs. your melee attack, but cover affects far more than melee attacks. I've always played it the same for other determinations of cover, but I really don't know what the intent is.