Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 3 / b / Solution

Using the normal vector from i, the normal line isFor $0<\theta <\pi$, the $y$- and $z$-coordinates vanish together exactly when $\lambda =-1$. At that point, the $x$-coordinate isThusAt $\theta =0$ or $\theta =\pi$, both the position vector and normal vector lie along the $x$-axis, so $L$ is the whole $x$-axis in either case.