= Solution
{c}
Using the <normal vector> from <3/a/i>, the <normal line> is
$$
\boxed{L:\ \mathbf x=\mathbf r(\theta,\phi)+\lambda(\sqrt2\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi),\quad\lambda\in\mathbb R.}
$$
For $0<\theta<\pi$, the $y$- and $z$-<coordinates> vanish together exactly when $\lambda=-1$. At that <point>, the $x$-<coordinate> is
$$
\frac{\cos\theta}{\sqrt2}-\sqrt2\cos\theta=-\frac{\cos\theta}{\sqrt2}.
$$
Thus
$$
\boxed{a=-\frac1{\sqrt2}.}
$$
At $\theta=0$ or $\theta=\pi$, both the <position vector> and <normal vector> lie along the $x$-axis, so \b[$L$ is the whole $x$-axis in either case].
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