The locus of a geometric figure can be obtained by encapsulating the known properties of the figure in some parametric equations, and then combining them into a single equation.
As I understand it, though, the equations aren't usually further examined to make sure that the locus curve and the figure are completely equivalent. That all the points of the initially specified geometric figure are present on the obtained locus is always true (obviously), but apparently it isn't required to check if we end up obtaining extra points that satisfy the final locus equation in some way, but aren't a part of the intended figure. Admittedly, I've never had cases where I got loci that had extraneous points, but it still seems necessary to check.
For example, a question requires me to find the locus of the centroid of the triangle formed by the varying points $$(\cos \alpha, \sin \alpha) \\(\sin \alpha, -\cos \alpha) \\ (1, 2), $$ and the parametric equations turn out to be $$3x-1=\sin\alpha+ \cos \alpha\\3y-2=\sin \alpha-\cos \alpha. $$ On squaring and adding them, we get the equation of a circle, $$(3x-1)^2+(3y-2)^2=2 \tag{*} $$
Here, all that I seem to have done is prove that a point satisfying the intended figure will necessarily satisfy ( * ), not that if a point satisfies ( * ), it will necessarily be part of the locus. I had to draw out the graphs of the trigonometric equations obtained from the R.H.S of the parametric equations to convince myself that both the sum and difference of $\cos$ and $\sin$ would only go so far as $\pm\sqrt{2} $, and that every possible $a$ and $b$ for which $a^2+b^2=2$ could be constructed by the various combinations (using signs) of the two, and that thus, the locus and the intended figure were equivalent.
Proving that the figure lies on the locus is easy, but proving the converse may take effort, maybe sometimes more than in this case. So is there really a reason to prove this equivalence, or is it always unnecessary?