All applicants need to apply in person. A 16-year-old child only requires one parents' consent (the parent must be listed on the birth certificate). If the kid has a previous passport, that can be used as proof of citizenship and a birth certificate is not necessary. See more info on travel.state.gov. (post by an employee)
Normally the minor child had to be present in order to get a passport. Also, the father has to sign an affidavit acknowledging that you're getting a passport otherwise he'll have too!
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