Past exam of the mathematical methods I course of the University of Oxford / 2015 / 10 / a / ii / Solution

The affine subspace is $S=a+\mathrm{span}(b,c)$. If $b,c$ are linearly independent, their linear span is all of ${\mathbb{R}}^2$, so $S$ is the whole plane. If they are proportional, $S$ is the line through $a$ in direction $b$. This line passes through the origin exactly when $a\in \mathrm{span}(b)$. Because $b$ is nonzero, the set cannot be a single point.