Shift of origin belongs to a broader concept in mathematics, known as transformation. By shifting an origin to a new point, the coordinates of each point change by a fixed amount.
Let be the new origin. Let be a point. Let the coordinates in the new (transformed) system be . Then,
I) Shift of origin changes the equation of locus.
II) The problems on shift of origin will generally ask for vanishing terms. All you have to do is, shift the origin, get the new equation of locus and equate the coefficients of vanishing terms to zero.