There are a lot more of these words. It’s quite surprising how many words trace back to reg-, especially words you’d never think of. For example…
Arrogant first showed up in the late fourteenth century, with arrogance also coming about around that time. Both words have Old French versions spelled exactly the same, and both come from the classical Latin arrogantem, proud, although arrogance comes via a noun form of the word (arrogantia). In Latin there’s also a verb form of the word, arrogare, which means to usurp or claim, a mix of the prefix ad-, to, and rogare, to ask. So it went from “to ask to” to usurp to… arrogant? It’s even more confusing considering rogare is from reg-, to move in a straight line. It’s thought to be figurative—I mean, obviously. It shouldn’t be too surprising that arrogate is from the same place. It showed up in the sixteenth century from the classical Latin arrogatus, which is also from arrogare.
Surrogate showed up in the early fifteenth century from the classical Latin surrogatus, substitute, from the verb subrogare (also spelled surrogare), to substitute or replace. Sub- means under or in the place of, and rogare is to ask, so it’s… to ask under? I guess they’re going for “to ask in the place of” here, which makes more sense than “to move in a straight line in place of” I guess.
Derogatory showed up in the sixteenth century, which means it might’ve showed up after forms we don’t use any more like derogate (early fifteenth century) and derogative (late fifteenth century). It comes to us through the Late Latin derogatorius, from the classical Latin derogatus and its verb form derogare, which means to detract from (or derogate). The de- means away here, so it’s “to ask away”. Or maybe “ask away from” since ask away has a different connotation these days. Asking away from detracts from someone’s credibility, I suppose.
Prerogative—which I’m just now learning is spelled “prero”—showed up in the late fourteenth century from the Old French prerogative and classical/Medieval Latin prerogative, all of which had roughly the same meaning as today. The verb form is praerogere, which meant something like to ask before others. The prae- is where pre- comes from and means before, so it’s to ask before.
Finally, interrogation showed up in the late fourteenth century, about a century before interrogate. It comes from the Old French interrogacion and classical Latin interrogationem, a question. Its verb form is interrogare, to ask, a mix of inter-, between, and of course rogare, to ask. To ask between. Well, it makes more sense than most of the origins today.