Translation:We are asking the woman about a book.
I think the answer is in the first discussion here. It depends on the usage of the verbs for "ask" in English and Greek. We use "ask" for information and for a request for help for instance. Whereas Greek has 2 different words depending on the kind of question. Am I right? English asks "for" help, but "about" information. Is this helping?
The hints are only weakly sentence-specific; the list often includes meanings that are not relevant to the current sentence but only to other sentences containing that word.
Sometimes, Greek uses the definite article (such as την) where English would use an indefinite article (such as "a"), especially with abstract nouns.
But this is not such a situation, and "a woman" is not an appropriate translation in this sentence.
I've read the comments above, and I understand that this sentence is a query for information as opposed to a request for an object. (Spanish is my native language and we also have two words for these.) What I don't understand is why "για" is used instead of "σχετικά με"? Is this one of those cases where that's just the way it is or is there a better explanation?