Irish Mathematical Olympiad

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.