The correct answer to the question is option 2: The Governor of the State.

When a Judge is appointed to a High Court in India, they are administered the oath of office by the Governor of the State. The Governor is an important constitutional authority in each state who represents the President of India at the state level. The Governor performs various functions and has certain powers, one of which is administering the oath of office and allegiance to the Judges of the High Court.

Option 1, The Chief Justice of the High Court, is not the correct answer as the Chief Justice does not administer the oath to other Judges. Their role is primarily to preside over the court proceedings and manage the administrative aspects of the High Court.

Option 3, The President of India, is also not the correct answer as it is the Governor and not the President who administers the oath to the Judges of the High Court.

Option 4, The Chief Justice of India, is not the correct answer either as they do not have the authority to administer the oath to Judges of the High Court. The Chief Justice of India has their own set of responsibilities within the Supreme Court.

Therefore, the correct answer is option 2: The Governor of the State.