Answer:
Although you didn't say which is the underlined verb, I assume it is prepare, since it is the only one in infinitive that would need an adjustment. So, The correct answer would be: By the time you have arrived, we will be preparing to go home.
Step-by-step explanation:
The Future Progressive Tense is the one that indicates a continuing action, something that will be happening or going on at a specific point in the future. It is formed with the modal "will" plus "be", plus the present participle of the verb (with an -ing ending). In this case, the specific point in the future would be By the time you have arrived, and the modal will, plus be, plus preparing (the present participle of prepare).