Passive skills (especially perception, but anything really), are representative of a sort of "always on" ability. So, for instance, if you're walking in a forest and there is a goblin ambush, you see them if your passive perception beats their stealth score. However, if you're specifically looking for ambushes, you'll roll it actively.
So in the secret doors case, when you walk into a room, if your passive perception is high enough to spot the secret door, you notice it. If you're specifically looking for a secret door, then you roll an active check to try to notice (naturally, if you roll less than 10, you don't learn anything new).
Generally, the only time to roll an active perception check is if you are looking for something that your passive perception did not pick up. Naturally, in the course of play this happens fairly regularly. But the passive skills are designed to reflect a characters natural competence and should be used when they are sufficient, or there is no particular rush.
Perception is for observation, Investigation is for deduction.
Some of this answer will be observations on how Wizards has done it so far and some of this will be logic, and some of it will be mechanics.
First for the headline question. It depends. when to use Investigation, and when to use Perception is not entirely clear yet, and I'm hoping we'll get more guidance in the DMG late this year. The guidance from the rules is that the two skills mostly seem to differ in the methods by which they are found.
Depending on the exercise, either, or both of the skills may be used.
For passive checks, you're almost always looking at Perception being the skill of record. While all skills can be used passively, some skills make more sense than others. Perception is the hallmark passive skill, whereas investigation makes less sense as a passive skill.
If the character is alert to the possibility of hidden objects/traps, but not actively searching, he's using passive perception.
The guidance for this seems to be (though we can't confirm yet), that the DC for actively looking for something is regularly about 5 less than the DC for passively looking for something (or sometimes actively looking always reveals it).
So here's how I would play it. Traps can be noticed with perception passively (usually DC 15). If the PC is looking, ask them how they are looking. If it's observational, then use Perception. If it's deductive, use Investigation. When they are searching for items, again, either skill is appropriate. This is somewhat counter to how WOTC has written adventures so far. They are always written to use perception to notice traps. Passive with a higher DC and active with a lower.
Investigation also has broader uses such as when you are trying to track clues, or put something together. It's also a great "roll for a hint" kind of skill if your PCs get stuck and need some help figuring out what to do next.
Best Answer
Yes, it's a pretty obvious secret door.
As you say, passive Wisdom (Perception) is 10 + WIS, so anyone with a non-negative WIS modifier will passively notice it.
But that doesn't mean you definitely see it.
The interesting exception I see is if your party is rushing through the area. Per Travel Pace moving 400'/minute or 4mi/hour will earn you a -5 to passive perception (PHB p.182). Lots of characters don't have the +5 WIS modifier necessary to offset that, and might blow right past it.
@Dale M points out the more likely scenario when this may happen: disadvantaged passive checks. By rule (PHB p.175), a passive score receives a -5 bonus when subject to disadvantage.1 Low light, obscuring conditions, or other environmental challenges could create disadvantage, similarly challenging one's ability to notice a DC10 secret door.
1 The "disadvantage=-5" rule of thumb is a lamentable addition to 5e culture. But, as this excellent answer shows, in the specific case of DC10 it's the perfect modifier. The second graph in the linked answer -- the vertical distance between the black and blue curves, to be precise -- shows this to be the case.