The Chief Justice of a High Court is appointed by the President of India in consultation with the Chief Justice of India and the Governor of the respective state. The correct answer is option C, as the Chief Minister of the respective state does not have a direct role in the appointment of the Chief Justice of a High Court.

Let's break down the process of appointment and understand the roles of the different authorities involved:

1. President of India:
The President of India is the head of state and exercises executive powers. In the case of appointing the Chief Justice of a High Court, the President plays a crucial role. The President appoints the Chief Justice of a High Court after consultation with the Chief Justice of India and the Governor of the respective state.

2. Chief Justice of India:
The Chief Justice of India is the head of the judiciary in the country. They play a significant role in the appointment of judges to the High Courts. The President consults the Chief Justice of India to seek their opinion and recommendation on the appointment of the Chief Justice of a High Court. The recommendation of the Chief Justice of India carries substantial weight in the final decision.

3. Governor of the respective state:
The Governor is the constitutional head of the state and represents the President at the state level. In the case of the appointment of the Chief Justice of a High Court, the Governor plays a vital role. The President consults the Governor of the respective state to gather information and insights about the candidate being considered for the position. The Governor's input is considered while making the final appointment.

It is important to note that the appointment of the Chief Justice of a High Court involves consultation with multiple authorities to ensure a fair and transparent process. The involvement of the Chief Justice of India and the Governor of the respective state helps in selecting a suitable candidate who possesses the necessary qualifications and experience for the position.

In conclusion, the President appoints the Chief Justice of a High Court in consultation with the Chief Justice of India and the Governor of the respective state. The Chief Minister of the respective state does not have a direct role in this process.