Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / ii / Solution

The recurrence $x_{n+2}=-{\lambda }^2{x}_n$ shows that every odd-indexed position vector is a scalar multiple of $x_1$, while every positive even-indexed position vector is a scalar multiple of $x_2$. HenceMoreover $x_2=\lambda a\times x_1$, so these two straight lines through the origin are perpendicular.