(a) The condition

\[ |z|=9 \]

means the points lie on a circle with centre \(O\) and radius \(9\).

The condition

\[ \frac12\pi\leq \arg z<\pi \]

restricts the locus to the part of the circle in the second quadrant.

So the locus is the arc of the circle \(|z|=9\) from \(9i\) to \(-9\), with \(9i\) included and \(-9\) not included.

(b) The transformation \(z\mapsto z^*\) reflects the locus in the real axis.

The transformation \(z^*\mapsto z^*+3\) then translates the reflected locus \(3\) units in the positive real direction.

So the locus is an arc of a circle with centre \(3\) and radius \(9\), from \(3-9i\) to \(-6\), with \(3-9i\) included and \(-6\) not included.