Hello there.
Though knowing perfectly the hold he had on this heart, and conscious he had only to speak and all the house would be obliged to bend to his wishes.
In this sentence, what does the word obliged mean?
Being indebted to someone is the correct answer. Since he had "a hold on this heart" and was aware of that hold, Heathcliff simply had to speak his will and the entire house felt compelled to answer his every desire.
Therefore answer would be,
being indebted to someone
Option (C)