hey carrot, sorry if you've already answered this but for parametric pt (iii) if you found QS=OS=a, is it justification enough to say Q is a variable point on fixed circle with centre S(0,a) and radius of a?
for locus of Q, is it ok to have worked out QS and OS as radii (which both ending up equal to a) and said it was part of a fixed circle through Q and O with centre S(0,a)? or did you have to use the slope of OQ?