= Solution
Let $L$ be the fixed line and choose $y\in L$. Take orthonormal vectors $e_1,e_2$ spanning the plane perpendicular to $L$. The process
$$
Y_t=\bigl(e_1\cdot(B_t-y),e_2\cdot(B_t-y)\bigr)
$$
is standard planar <Brownian motion> starting from $Y_0\ne0$. Indeed the projected increments are centered <Gaussian random vectors> with covariance $(t-s)I_2$, are <independent> on disjoint intervals, and have continuous paths.
The event $B_t\in L$ is exactly $Y_t=0$. The <polar point for planar Brownian motion> result in (b) therefore gives
$$
\boxed{\mathbb P_x(\exists t\geq0:B_t\in L)=0\qquad(x\notin L).}
$$
This is a <projection criterion for Brownian avoidance of affine subspaces>: avoiding a fixed point in a two-dimensional projection excludes hitting the original line.
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