\((0, k)\) is the point to which the origin is to be shifted by the translation of the axes so as to remove the first degree terms from the equation \(a x^2-2 x y+b y^2-2 x+4 y+1=0\) and \(\frac{1}{2} \tan ^{-1}(2)\) is the angle through which the coordinate axes are to be rotated about the origin to remove the \(x y\) term from the given equation, then \(a+b=\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.