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According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix B)

My question is if I add a linear constraint to the problem will this affect the strong duality between the relaxed problem and the original problem ?

Thank you

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Your problem is a special case of the extended CDT subproblem. You can check whether you can obtain its solution using its dual problem's solution by checking the necessary and sufficient conditions for strong duality in your case.

Here is a paper for your reference:

[Strong Duality for the CDT Subproblem: A Necessary and Sufficient Condition][1]

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