The Judge of the High Court is appointed by:-

Article 217(1) of the Constitution of India vests the power to appoint every Judge of a High Court — including the Chief Justice — in the President of India, who acts by warrant under hand and seal after consultation with the Chief Justice of India, the Governor of the State concerned, and (for a judge other than the Chief Justice) the Chief Justice of that High Court.

Applying this to the question: the appointing authority named in the Constitution is the President, not any state or union executive officeholder. The Home Minister and the Chief Minister exercise no role in this appointment process at all, and neither does the Prime Minister — Article 217 names only the Chief Justice of India, the State Governor, and (where relevant) the Chief Justice of the High Court concerned as the consultees; the constitutional appointing authority itself remains the President.

This design is consistent with Article 124, which similarly makes the President the appointing authority for Supreme Court Judges, again after consultation with named judicial authorities. Both provisions place the formal appointing power in the Head of State — confirming the President as the correct answer among the options offered.