What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
If possible, show some reference please?
Mathematically, they both are the same curve but they arise from slightly different but related problems.
Both problems are solved via Variational Calculus.
Here an illustration of the Tautochrone from Wikipedia (by Claudio Rocchini):
By comparison, this is the problem you are trying to solve with a Brachistochrone (Maxim Razin on Wikipedia):