Irska

Let $R$ be the foot of the perpendicular from $O$ on the given line and let $S$ be chosen on $OR$ such that $|OR|\cdot |OS|=|OP|\cdot |OQ|$, i.e. $S$ is a point on the locus. The value of the constant determines the position of $S$, either between $O$ and $R$ or not. Then $P,Q,S$ and $R$ lie on a circle in all cases (see formulation of the hint). Hence $\angle PQS=90^{\circ }$ since $\angle PRO=90^{\circ }$. This implies that $Q$ lies on the circumference of the circle on $OS$ as diameter which is a fixed circle. In every case $Q$ can be taken on either side of $O$ on the line $OP$. The analysis in every case (see drawings below) is similar to above.