Question

she asked both of us about borrowing your telescope for a week. which pronoun is a third person personal pronoun?
Your
Us
Both
She

Answer 1

((She)) is the answer! 
The other ones are first person.

Answer 2

There are a couple of pronouns in this sentence - she, us, both, and your. However, only one of these is a third person personal pronoun, and that is she.
Us is first person plural, both also refers to first person plural, and your is second person singular.